Muestra del libro Mathematics 3º ESO Proyecto 5 etapas

José María Arias Cabezas Ildefonso Maza Sáez

A guide to your book

With the book you are holding you will:

➜ Learn Maths using the 5-stage methodology: Engage, Explore, Explain, Elaborate and Evaluate, the same method used in the Spanish book.

➜ Consolidate and improve your English proficiency through activities where you will practise both written and oral comprehension and expression.

The following information is given:

1. The content of the module and the 5 stages:

➜ Engage: a stimulating GeoGebra applet related to the core concepts covered in the block.

➜ Explore: an interactive and dynamic activity based on the applet that lets you explore your current knowledge and relate it to the concepts you will be studying.

➜ Elaborate: a section to investigate and apply the concepts.

2. A learning situation with two phases:

➜ Explore: searching for information on a concept to be worked on.

➜ Elaborate: a short writing activity to present the information obtained.

Main content

Each unit’s double page contains the Explain stage where the core concepts are presented. This is supported by a solved exercise or problem to illustrate the mathematical concept. The exercise is solved with straightforward operations so the students can focus on the new content being covered. The final Elaborate section has exercises and problems to enhance and consolidate the acquired knowledge.

Learning experience

A comprehensive learning experience will enable you to develop all the key competences and core concepts you have acquired, as well as enhance your teamwork skills. Additionally, you will be able to communicate the results in a creative manner, using different formats.

Evaluation

The last step in the 5-stage methodology is Evaluate, which has two parts:

➜ A solved evaluation that serves as a review of the content covered.

➜ A proposed evaluation covering the module’s content.

Core concepts

1 Rational and irrational numbers

2 Powers and roots

3 Arithmetic problems

4 Sequences and progressions

e e e e

Open the applet using the QR.

1 Observe the solved exercise and work out the proposed exercise.

2 You can investigate further with other polygonal numbers.

Learning experience

At the end of this module, you will find a comprehensive activity that will allow you to work in teams. You will need to apply all the core concepts you have acquired. In addition, you will be able to communicate the results in a creative manner, using different formats.

e e e e

1 Find information on different energy sources and their CO2 emissions.

2 Write a few sentences with the information you have obtained.

UNIT 1 Rational and irrational numbers

How do we work with fractions? 1

Setting your calculator

To display improper fractions directly:

MODE (Disp) 1 (d/c) 2

To use decimal notation:

MODE (Disp) 1 ▶ (Comma) 2

Opposite fractions

The opposite fraction of a fraction is the one you get when you use the opposite sign.

The opposite of 3 4 is –3 4

The opposite of –2 5 is 2 5

How do we add and subtract fractions?

To find the sum or difference of fractions with the same denominator:

• Numerators: add or subtract the numerators.

• Denominators: keep the same denominator.

To find the sum or difference of fractions with different denominators:

• Numerators: find the least common multiple (LCM) of the denominators. Divide the LCM by each denominator. Multiply each numerator by the corresponding quotient. Add or subtract the numerators.

• Denominators: the LCM of the denominators.

After completing the steps above, find the irreducible fraction, in other words, divide the numerator and the denominator by their greatest common divisor (GCD).

Reciprocal fractions

The reciprocal fraction of a fraction is the one you get when the numerator and denominator switch places.

The reciprocal of 2 3 is 3 2

The reciprocal of –4 5 is –5 4

How do we multiply fractions?

To find the product of two fractions:

• Numerators: multiply the numerators.

• Denominators: multiply the denominators.

How do we divide fractions?

To divide fractions, multiply the first fraction by the reciprocal of the second fraction.

KCF rule: Keep, Change, Flip

Keep the first fraction.

Change division to multiplication.

Flip the second fraction.

What is the order of operations?

When an expression with fractions has multiple operations, they must be performed in this order:

a) Brackets

b) Exponents and roots

c) Multiplication and division

d) Addition and subtraction

Do not forget the sign rule when multiplying or dividing.

1. Calculate mentally.

4. Divide the fractions.

If the operations are on the same level (for example, an operation on level c contains only addition and subtraction) and are multiplication with division or division with division, you work from left to right.

3. Multiply the fractions. a)

6. A lorry can hold 12 000 kg. It is 3/5 loaded. How many kilos is it carrying?

7. One-sixth of the water is removed from a water tank with a capacity of 1 500 litres. Then 750 litres are removed. What fraction remains?

How do we use real numbers? 2

What are real numbers?

Irrational numbers

Irrational numbers are numbers that cannot be expressed as the quotient of two integers. When written in decimal form they are not exact and do not repeat.

The quotient of the length of a circumference and the diameter of the circle it forms is:

π = 3.1415926535914039… π = 3.141592654

If the root of a natural number is not exact, it is an irrational number.

1 For a square with sides measuring 1 cm, find the length of a diagonal.

√ 2 = 1.4142135623715… √–2 = 1.414213562

2 For a cube with a volume of 5 cm3, find the length of an edge.

3 √ 5 = 1.709975947 3 √–5 = 1.709975947

Graphic representation

To graphically represent irrational numbers that are square roots, use the Pythagorean theorem. You can also use an approximation of its decimal form.

Real numbers

The set of real numbers is comprised of rational and irrational numbers.

How do we express real numbers?

Approximation

Decimals numbers can be expressed exactly or as approximations. When you round a number, you shorten it. Sometimes you round up to the closest number and sometimes you round down

The significant figures are the digits used to express an approximated number.

There are two ways to approximate numbers:

• Rounding : If the last insignificant figure is 0, 1, 2, 3, or 4, you do not change the digit to its left. If the last insignificant figure is 5, 6, 7, 8, or 9, you add one to the digit to its left.

• Truncation: The insignificant figures are removed.

Absolute and relative error

• An absolute error is the difference (which is always absolute, meaning a positive figure) between the exact value and the approximation.

• A relative error is the quotient of the absolute error and the exact value.

3 Find the absolute error and relative error for rounding the number √ 5 to two decimal places.

If you round √ 5 = 2.236067… to 2.24, the errors are:

• Absolute error = | √ 5 – 2.24| = |2.236067… – 2.24| = 0.003932…

• Relative error = 0.003932… 2.236067… = 0.00176

8. Classify the numbers as rational or irrational 2 3 π – 7 √ 3 1 2 5 √ 7

9. Graphically represent the following irrational numbers.

a) √ 5

b) √ 6

10. Round to two decimal places and solve the operation.

a) 3.456 + 0.342 – 2.108

b) 15.362 · 3.236

c) 45.875 : 3.236

d) 2.458 + 42.253 : 8.417

These numbers are rounded to two decimal places:

a) 4.52385 = 4.52

b) 3.34813 = 3.35

c) 0.62543 = 0.63

d) 18.59732 = 18.60

These numbers are truncated to two decimal places:

a) 4.52385 = 4.52

b) 3.34813 = 3.34

c) 0.62543 = 0.62

d) 18.59732 = 18.59

Calculator

To round the quotient of 8/3 to two decimal places with a calculator:

MODE (Fix) 1

(Fix 0 9?) 2

8 ÷ 3 = 2.67

To cancel rounding:

MODE (Norm) 3

(Norm 1 2?) 1

11. Calculate the absolute error for rounding the following numbers to two decimal places.

a) 3.1415

b) 0.0278

c) 1.2068 d) 5.3975

12. Round the numbers to the decimal place indicated.

a) 4.3248 to the second decimal place.

b) 58.15 to the whole number.

c) 0.00482 to the third decimal place.

d) 37.4932 to the first decimal place.

13. What are the absolute and relative errors if you round the width of a shelf measuring 345 cm to 3.5 m?

14. José and Sonia had to round some numbers to complete an exercise. José approximated a real distance of 20 m as 19 m. Sonia approximated a real distance of 50 m as 48 m. Who has the largest error?

UNIT 2 Powers and roots

a) 53 = 5 · 5 · 5 = 125

b) ( 2 5 )3 = 2 5 · 2 5 · 2 5 = 8 125

What are powers? 1

What is a power with a natural exponent?

A power is the product of equal factors.

Exponent

n-times

Base a n = a · a · … · a

In other words, it is a number multiplied by itself. The base is the number that is multiplied. The exponent is the number of times it is multiplied.

The sign of a power

Special cases

a) Any number other than zero raised to the zero power is equal to one.

a0 = 1, a ≠ 0

30 = 1

b) Any number raised to the first power is equal to itself.

a1 = a 51 = 5

The sign of a power is positive unless its base is negative and the value of the exponent is odd, in which case it is negative.

What are the properties of powers?

a) Products of powers with the same base

The product of two powers with the same base is another power with the same base, and the new exponent is the sum of the two exponents.

23 · 24 = 27

a n · a p = an + p

b) Quotients of powers with the same base

The quotient of two powers with the same base is another power with the same base, and the new exponent is the difference between the two exponents.

a n : a p = an – p

c) Powers of powers

75 : 73 = 72

The power of a power is another power with the same base and the new exponent is the product of the exponents.

(a n)p = a n · p

(72)3 = 76

What is a power with an integer exponent?

A power with a negative whole number exponent is equal to one divided by the power, but with a positive exponent.

a – n = 1 an when a ≠ 0

Scientific notation

Scientific notation is a method for expressing very large and very small numbers. They are written as the product of a decimal number and a power of ten. Only one digit is to the left of the decimal point, and it must be between 1 and 9

How can we avoid commom mistakes?

Do not confuse (– a)n with – an

If n is odd, they are equal.

If n is even, (– a)n equals an

1. Calculate.

a) 34 b) (– 3)4 c) 35 d) (– 3)5

2. Apply the properties of powers. Express the result as a single power.

a) 25 · 24 b) 59 : 53 c) (24)3 d) 32 · 33 · 34

3. Multiply to eliminate the brackets.

a) 3a 2b(2ab 2 – 5a 2b 3)

b) 2x 3y 2z(3xy 2z 2 + 4x 2yz 3 – 6x 3z 4)

4. Remove all common factors.

a) 6a 3b 2 – 8a 4b 5

b) 18x 2y 5z 2 + 12x 2y 3z 3 – 6x 3y 3z 4

5. Convert the powers to fractions mentally.

a) 2– 1

b) (– 2)– 2

c) 2– 3

d) (– 2)– 3

6. Write the numbers in scientific notation:

a) 54 689 000 000 000 000

b) One ten-millionth of four units.

7. Our solar system is 27 700 light years from the centre of our galaxy. Express the distance in kilometres and in scientific notation. A light year is the distance light travels in a year at the speed of 300 000 km/s

√ Radical sign

n Index

a Radicand

b Root

How do we work with radicals? 2

What are radicals?

The nth root of any number a is another number b such that b raised to the power of n is a: n √a = b if bn = a

An nth root operation is the opposite of exponentiation.

If the index of a radical is an odd number, then the radical has only one root. If the index is even and the radicand is positive, then the radical has two roots, one positive and one negative. If the radicand is negative, then the radical has no real number root.

Order of operation

a) Brackets

b) Exponents and roots

c) Multiplication and division

d) Addition and subtraction (

(53 – 119)

– 119)

2 = = 6 · 2 = 12

Convention

In combined operations with radicals with even numbered indexes, the root takes the sign of the radical. √

3 √8 = 2 since 23 = 8

Simplifying radicals

4 √81 = ±3 since ⎧ ⎨ ⎩ 34 = 81 (– 3)4 = 81

To simplify a radical, divide the index of the root and the exponent of the radicand by their greatest common divisor (GCD). This type of simplification is used when both the index and radicand have radicals.

18√512 = 18 : 6 √512 : 6 = 3 √5 2

GCD(12, 18) = 6

Introducing factors into the radicand

To introduce a factor into a radical, raise the factor to the same power as the radical’s index and multiply. 2

Removing factors from the radicand

To remove factors from the radicand, first break down the radicand into its prime factors. Then divide the exponent of each factor by the index. The quotients are moved outside the radical sign as each term’s exponent. The remainders of divisions remain inside the radical sign as each term’s exponent. All exponents of the radicand should be less than the index of the radical.

How do we add and subtract radicals?

Like radicals are radicals that have the same index and the same radicand after they are simplifaied.

To add and subtract radicals, they must be like radicals. If they are like radicals, you can simply add or subtract the coefficients. The radical remains the same.

Mental calculations

To remove factors from a square radical, break down the radicand into its products and find the largest perfect square and a number.

√ 50 = √ 25 · 2 = 5 √ 2

√ 8 = √ 4 · 2 = 2 √ 2

√ 18 = √ 9 · 2 = 3 √ 2

What are the properties of powers and radicals?

Powers Example

a n · a p = a n + p 53 · 54 = 57

a n : a p = a n – p 28 : 23 = 25

(a n)p = a n · p (53)2 = 56

(a · b)n = a n · b n (2 · 3)5 = 25 · 35

(a : b)n = a n : b n (5 : 7)3 = 53 : 73

a – n = 1 a n 2 – 3 = 1 23

a 1/n = n √ a 61/5 = 5 √ 6

a – 1/n = 1 n √ a 5 – 1/4 = 1 4 √ 5

a p/n = n √ a p 72/3 = 3 √ 72

a – p/n = 1 n √ a p 6 – 3/4 = 1 4 √ 63

8. Calculate mentally if possible.

Radicals Example

ns√ a ps = n √ a p 10√ 76 = 5 · 2 √ 73·2 = 5 √ 73

a n √ b = n √ a n · b 2 3 √ 5 = 3 √ 23 · 5 = 3 √ 40

n √ a n √ b = n √ a · b 5 √ 2 · 5 √ 3 = 5 √ 6

n √ a : n √ b = n √ a : b 3 √ 2 : 3 √ 5 = 3 √ 2 : 5

( n √ a )p = n √ a p ( 3 √ 7 )2 = 3 √ 72

n √ p √–a = n p √ a 5 √ 3 √–7 = 15√ 7

n √ a = a 1/n 3 √ 7 = 71/3

1

n √ a = a – 1/n 1 5 √ 2 = 2– 1/5

n √ a p = a p/n 3 √ 52 = 52/3 1

n

a) a) √ 25 b) 3 √ – 125 c) √ – 49 d) 3 √ – 27

9. Simplify the radicals.

a) a) 6 √ 54 b) 9 √ 56 c) 12√ 58 d) 24√ 518

10. Calculate the radicals by factoring the radicand.

a) √ 32 400 b) 3 √ 3 375 c) 5 √ 1 024

11. Extract the factors if possible.

a) √ 81a 5 bc 6 b) 3 √ 128a 8 b 2 c 15

12. Add and subtract the radicals.

a) √ 50 – √ 32 + √ 18 b) 5 √ 98 – 3 √ 200 + 4 √ 8

13. Apply the properties of radicals to express the radicals with a single root.

a) √ 5 · √ 3 b) √ 6 : √ 3 c) ( 3 √ 5 )2 d) 3 √ √ 5

14. Write the powers as radicals and solve.

a) 271/3 b) 49 – 1/2

c) 1283/7 d) 243 – 2/5

UNIT 3 Arithmetic problems

What are simple proportions? 1

How do we solve direct proportionality problems?

Two quantities are considered to be directly proportional if the quotient of the quantities is constant.

The direct proportionality constant is calculated by dividing any element of the second quantity by the corresponding element in the first quantity.

1 The number of kilos of apples and the value of the apples are directly proportional. The direct proportionality constant is:

Problem solving: direct rule of three

Unitary method

Direct proportionality problems can also be solved using the unitary method, in other words, by calculating the value of a single unit.

a) Find the value of a single unit.

If 7 kg cost €14.70, then 1 kg costs:

b) Calculate the unknown value.

The rule of three method or cross-multiplication is a method for finding the fourth quantity in a direct proportion.

a) Identify the quantities and their units.

b) Put the quantities in order with the unknown amount in the last place.

c) Determine if the proportion is direct. In direct proportions, if one quantity increases or decreases, the second variable also increases (+ and +) or decreases (- and -).

d) Find the proportion and calculate the fourth quantity.

2 If 7 kg of apples cost €14.20, how much will 5 kg cost?

• The unknown quantity is money (€). This goes in the last position.

• The proportion is direct because as the number of kilos increases, the amount of money increases in the same way, from + to +

How do we solve inverse proportionality problems?

Two quantities are inversely proportional when their product is constant.

The inverse proportionality constant is the value of the constant product.

Problem solving: inverse rule of three

Apply the method below.

e) Identify the quantities and their units.

f) Put the quantities in order with the unknown amount in the last place.

g) Determine if the proportion is inverse. In an inverse proportion, if one quantity increases, the other quantity must decrease (+ and -). Alternatively, if one quantity decreases, the other must increase (- and +).

h) When the relationship of the quantities is inversely proportional, invert the ratio for quantity A and calculate.

At a speed of 10 km/h, a certain distance is travelled in 6 hours.

3 A farmer has enough hay to feed 15 cows for 8 days. If the farmer buys 5 more cows, how many days can she feed her cows for?

The unknown quantity is time (days). This is put in last place. The proportion is inverse since, as the numbers of cows increases, the number of days decreases (from + to -).

1. If 8 video games cost €212, how many video games can you buy for €371 ?

2. A 15 m pipe weighs 210 kg. Another pipe of the same material and diameter weighs 308 kg. How long is this pipe?

3. Nine identical light bulbs consume 54 kWh. If 15 light bulbs are used in the same conditions, how many kWh will be consumed?

The quantities speed and time are inversely proportional.

The inverse proportionality constant is:

10 · 6 = 20 · 3 = … = 60

Unitary method

Indirect proportionality problems can also be solved using the unitary method. Find the value of one unit.

If 15 cows have food for 8 days, 1 cow has food for:

15 · 8 = 120 days

Calculate the value. Twenty cows will have food for:

120 20 = 6 days

4. Four friends pay equal parts of the rent for a summer apartment. Each one pays €375. If two more friends join them, how much will they each pay?

5. A car travels for 1 and half hours at 65 km/h. What speed must the car travel to cover the same distance in 75 minutes?

6. Three open taps fill a container in 20 hours. How long will it take to fill the same container with five open taps?

What are compound proportions? 2

How do we solve problems with compound proportions?

Proportions involving two or more quantities are called compound proportions.

Problem solving: compound rule of three

To solve problems using the compound rule of three method follow the steps below.

a) Identify the quantities and their units.

b) Put the quantities in order with the unknown quantity in the last place.

c) Determine what kind of proportion there is between the known and unknown quantities.

Direct or inverse proportion?

Direct or inverse proportion?

d) Set up the proportion as a direct or inverse proportion, considering if the quantities are directly or inversely proportional, and solve.

4 Two workers take 10 days to install 100 m of pipes. How long will it take 5 workers to install 350 metres of pipes?

The unknown quantity is time (days). This is put in last place.

Exercise number 4 has different proportional quantities: the number of workers, the length of the pipes, and time.

The first goes from + to –(inverse).

The second goes from + to + (direct).

How do we solve problems with simple interest?

Interest is money earned on capital deposited in a financial institution.

Simple interest is interest computed and earned only on the original capital, not on subsequent interest earned from the original capital.

Interest is computed using the original amount of capital deposited, and its yield or percentage.

The compound rule of three can be used to calculate interest.

Practical rule: Use this formula to calculate interest.

The final amount of capital will be:

5 You deposit €5 000 in an account that earns 4% simple annual interest for two years. What will the final amount of capital be?

The interest will be:

The

Notice

If the time an amount of money is deposited is less than a year, a proportional part of the interest is earned. If t is in quarters: I = c · r · t 4

If t is in months: I = c · r · t 12

If t is in days: I = c · r · t 360

5. Six workers take 30 days in order to install 150 m of pipe. Calculate how many metres 14 workers can install in 24 days.

6. The cost of feeding 135 people is €2 250 a day. Calculate how many people can be fed for €90 days with €12 000

7. A person reads 2 hours a day at a rate of 5 pages an hour and takes 15 days to read a book. If the person reads 3 hours a day at a rate of 8 pages an hour, how many days will it take him to read the same book?

8. Calculate the interest earned on initial capital of €9 000 at 1.5% interest in 3 years.

9. How much capital should be deposited at 2.5% interest to produce €400 in 2 years?

10. What should the interest rate be for an initial capital deposit of €6 500 in order to earn €526.50 in 18 months?

11. How many months will €25 000 of initial capital need to remain deposited at an interest rate of 1.5% to earn €562.50 in interest?

What are proportional distribution and percentages?

How do we solve problems with proportional distribution?

Directly proportional distribution

Use this procedure to distribute a quantity N in parts directly proportional to quantities a, b, c, etc.

a) Calculate constant k , in other words, the part of N that corresponds to each unit of the total of the known quantities a, b, c, etc. Use this formula:

k = N a + b + c

b) With unit value k, calculate the unknown values.

6 Divide €3 000 in directly proportional shares of 4, 7 and 9

a) Calculate the amount of money that corresponds to one unit.

3000

4 + 7 + 9 = €150

b) Calculate how much money each share receives.

x = 150 · 4 = €600

y = 150 · 7 = €1050

z = 150 · 9 = €1350

Inversely proportional distribution

To distribute quantity N in parts that are inversely proportional to known quantities a , b , c , etc., divide the money as you would using the directly proportional method above, but using reciprocal fractions 1/a, 1/b, 1/c, etc.

a) Write reciprocal fractions a, b, c, etc. Reduce to the common denominator if necessary.

b) Calculate the amounts for each part using the nominators.

7 Distribute €4 200 in inversely proportional shares of 3, 5 and 6

a) Write reciprocal fractions and reduce to the common denominator if necessary:

LCM(3, 5, 6) = 30 ⇒ 1 3 = 10 30 , 1 5 = 6 30 , 1 6 = 5 30

b) Divide the money proportionally using 10, 6 and 5

Each unit is worth: 4200 10 + 6 + 5 = €200

x = 200 · 10 = €2 000

y = 200 · 6 = €1 200

z = 200 · 5 = €1 000

How do we solve problems with percentages?

• A percentage decrease of a quantity is the amount it decreases expressed in percentage.

• A percentage increase of a quantity is the amount it increases expressed in percentage.

8 A shop is offering a 15% discount on a mobile phone that costs € 700. What price would you pay if you bought it?

F = I · P ⇒ F = 700 · 0.85 = €595

700 × 0.85 = 595

9 A television costs €605, which includes 21% value-added tax (VAT). How much does it cost without VAT?

I = F P ⇒ I = 605 : 1.21 = €500

605 ÷ 1.21 = 500

10 A pair of shoes is €85 and you buy it on sale for €68. What percentage was the discount?

P = F I ⇒ P = 68 : 85 = 0.8 ⇒ you pay 80% ⇒

the discount applied is 20%

68 ÷ 85 = 0.8

Sequential increases and decreases with percentages

To calculate sequential increases and decreases with percentages, find the total percentage change by multiplying the percentage change in each step.

11 The price of a computer component that cost €100 last year has increased by 10%. Because you are buying it this year, you receive a discount of 10%. How much do you pay for it?

Percentage increase: 1 + 0.1 = 1.1

Percentage decrease: 1 – 0.1 = 0.9

Total percentage change: 1.1 · 0.9 = 0.99. You pay 99% of the price, which is100 · 0.99 = €99 Magic

14. Divide €15 000 in directly proportional parts of 2, 3 and 5

15. Divide €11 050 in inversely proportional parts of 2, 3 and 4

16. A worker is paid €1 440 each month. Five percent is taken from this amount to pay for insurance. How much is deducted from her wages?

17. A repair shop charges 21% VAT on a €168 bill. What is the final amount?

F = Final

I

The

Percentage change

• Percent decrease.

Reduction of 15%

Percentage change:

1 – 0.15 = 0.85

• Percent increase.

Increase of 21%

Percentage change:

1 + 0.21 = 1.21

18. In a 500 g coffee mixture, 100 g is roasted coffee and the rest is not roasted. What percentage is roasted?

19. A 20% discount is applied to an invoice of €350 and 21% VAT is charged as well. Calculate the final amount you have to pay.

20. You buy a television that has a 20% discount and you have to pay 21% VAT. If you pay €232, what was the original price of the television?

UNIT 4 Sequences and progressions

What is a sequence? 1

Plotting a sequence on a graph

A sequence can be interpreted as a discrete function

What is a sequence of real numbers?

A sequence of real numbers is a set of real numbers ordered in a sequence, in other words, each number in the sequence is in a specific position.

The numbers 2, 4, 6, 8, 10, 12, etc. are a set of numbers that are ordered in a sequence.

123579 046810

Polygonal number or figurate numbers

A polygonal number or figurate number is a natural number that can be represented by a set of equidistant points forming a regular geometric figure.

A triangular figure with 8 dots is:

Note that the numbers are even numbers.

The rule they follow is ‘to obtain the next number in the sequence, 2 is added to the previous number’.

The sequence can be continued indefinitely.

Terms of a sequence

The terms of a sequence are each of the numbers in it.

Each term in a sequence occupies a specific position. To represent a term, a letter is used with a numerical subscript which represents its position (for example, a3).

Avoiding common errors

Do not confuse the subscript that accompanies a letter a1, a2, a3, etc. (which indicates the position of the term) with the value of the term.

The sequence {3, 6, 19, 15, 21, …} is represented as:

a1, a2, a3, a4, a5 where:

a1 = 3

a2 = 6

a3 = 10

a4 = 15

→ Means the number 3 occupies position 1 in the sequence.

→ Means the number 6 occupies position 2 in the sequence.

→ Means the number 10 occupies position 3 in the sequence.

→ Means the number 15 occupies position 4 in the sequence.

1, 3, 6, 10, 15, 21, 28, 36

a5 = 21

→ Means the number 21 occupies position 5 in the sequence, etc.

What is a regular sequence?

A sequence is considered regular when its terms follow a specific rule.

A regular sequence can be determined in different ways:

With the first terms in the sequence

If you have the first three or four numbers in a sequence, you can try to deduce the formula for the sequence.

With the formula to create the sequence

If you have the formula for the sequence, you can create the sequence based on the formula.

1 Find the sequence of perfect squares of natural numbers. The sequence is: 1, 4, 9, 16, 25, 36, 49, 64…

With the general term of the sequence

This is the formula that you can use to find any term in the sequence. This is generally how sequences of real numbers are represented in mathematics.

General term of the sequence

The general term of the sequence is a formula that relates the position n that each term occupies to its corresponding value. It is written as a n

With the general term, any term in the sequence can be calculated by substituting the letter n for its position in the series.

2 Find the first three terms in the sequence a n = 5n + 1

a1 = 5 · 1 + 1 = 6

a2 = 5 · 2 + 1 = 11

a3 = 5 · 3 + 1 = 16

1. Find the first ten terms in the sequences below:

a) 3, 8, 13, 18… b) 8, 4, 0, –4…

c) 2, –2, 2, –2… d) 1/2, 1/4, 1/6, 1/8…

2. Find the first ten terms in the sequences below:

a) 2, 1, 2, 4, 2, 7… b) 1, 1, 2, 3, 5, 8…

c) 2, 1, 4, 3, 6, 5… d) 1, –2, 4, –8…

In the sequence {2, 5, 8, 11, …} note that each term is obtained by adding 3 to the previous term.

The sequence can be continued: a5 = 14, a6 = 17, a7 = 20

3. Calculate the first terms in the sequences below:

a) a n = 3n + 2 b) a n = (n + 1)2

c) a n = 3 · 2n d) a n = (–2)n

4. Find the first four positive terms in the sequences below. Try to determine the general term mentally:

a) Even numbers

b) Odd numbers

c) Multiples of 5 d) Perfect cubes

What is an arithmetic progression? 2

What are the properties of an arithmetic progression?

What is the first natural number?

Zero is the first natural number. However, sometimes zero is not included in the sequence of natural numbers because it is more logical to have one in the first position, two in the second position, etc.

An arithmetic progression is a numerical progression in which each successive term is determined by adding the previous term to a constant called the common difference. The letter d is generally used to represent the common difference.

The common difference d in an arithmetic progression can be calculated by subtracting two consecutive terms.

The sequence {3, 7, 11, 15, …} is an arithmetic sequence.

d = a2 – a1 = 7 – 3 = 4; d = a3 – a2 = 11 – 7 = 4; d = a4 – a3 = 15 – 11 = 4

The common difference is d = 4. Each term is obtained by adding 4 to the previous term.

Newer calculators

Older calculators

3 = Ans + 4 = = = 4 + + 3 = =

Terms of the progression

a1 = 3 d = 4

a2 = a1 + d = 3 + 4 = 7

a3 = a2 + d = 7 + 4 = 11 …

What is the general term of an arithmetic progression?

The general term of an arithmetic progression is:

a n = a1 + (n – 1) d

3 For the series {2, 5, 8, 11, …}, calculate the general term.

a n = a1 + (n – 1) d

a1 = 2; d = 3

a

n = 2 + (n – 1) · 3

a

n = 2 + 3n – 3

a n = 3n – 1

Note

Every term in an arithmetic progression is equal to the first term plus the difference between the terms multiplied by the number of terms before it. So, for general term an, it would be the first term plus n – 1 multiplied by the difference between terms.

In an arithmetic progression, you can verify that:

a n = ak + (n – k )d

4 The following terms in an arithmetic progression are known. a3 = 11 and a9 = 29. Calculate the common difference.

a n = ak + (n – k )d ⇒ a9 = a3 + (9 – 3)d

Adding terms in an arithmetic progression

The sum of the first n terms in an arithmetic progression is represented by S n and is:

S n = a1 + a n 2 · n

5 For the arithmetic progression with general term a n = 7 n – 5, calculate the sum of the first 25 terms.

S n = a1 + a n 2 · n ⇒ S25 = a1 + a25 2 · 25

Calculate a1 and a25 :

5. Find the general term for the following arithmetic progression:

a) 5, 9, 13, 17… b) 6, 3, 0, –3… c) 2/3, 1/3, 0, –1/3… d) 1/2, 1, 3/2, 2…

6. For the arithmetic progression with first term a 1 = 6 and d = 2.5, find the general term and the first three terms.

7. In the progression {5, 9, 13, 17, …}, which term has a value of 49?

8. Terms a5 = 19 and a8 = 28 are known in an arithmetic progression. Calculate the common difference and the first term.

9. Calculate the sum of the first 25 terms of the arithmetic progression with the general term below:

a n = 2n + 6

10. Calculate the sum of the first 12 terms of the arithmetic progression with the general term below:

a n = 3n 2 + 2

What is a geometric progression? 3

The progression (3, 12, 48, 192, …) is geometric.

r = a2 a1 = 12 3 = 4

r = a3 a2 = 48 12 = 4

r = a4 a3 = 192 48 = 4

The common ratio is r = 4 and each term in the progression is obtained by multiplying the previous term by 4

Terms in the progression:

a1 = 3

a2 = a1 · r = 3 · 4 = 12

a3 = a2 · r = 12 · 4 = 48

a4 = a3 · r = 48 · 4 = 192 …

What are the properties of a geometric progression?

A geometric progression is a progression in which each term is obtained by multiplying the previous term by a constant called the common ratio. The letter r is generally used to represent the common ratio.

The common ratio r of a geometric series is calculated by dividing consecutive terms.

The progression {3, 12, 48, 192, …} is geometric. r = a2 a1 = 12 3 = 4 ⇒ r = a3 a2 = 48 12 = 4 ⇒ r = a4 a3 = 192 48 = 4

The common ratio is r = 4 and each term in the progression is obtained by multiplying the previous term by 4

Terms in the progression:

a1 = 3 ⇒ a2 = a1 · r = 3 · 4 = 12 ⇒ a3 = a2 · r = 12 · 4 = 48 ⇒ a4 = a3 · r = 48 · 4 = 192 …

General term of a geometric progression

The general term of a geometric progression is:

a n = a1 · r n – 1

6 Given the progression {3, 6, 12, 24, …}, calculate the general term.

a n = a1 · r n – 1; a1 = 3

The common ratio is: r = a2 a1 = 6 3 = 2

The general term is: a n = a1 · r n – 1 ⇒ a n = 3 · 2n – 1

In a geometric progression, you can verify that:

a n = ak · r n – k

With a calculator

Newer calculators:

3 = Ans ×

4 = = …

Older calculators:

4 × × 3 = = …

7 For a geometric progression a3 = 18 and a7 = 1 458, calculate the common ratio.

a n = ak · r n – k ⇒ a7 = a3 · r 7 – 3 ⇒ 1 458 = 18r 4 ⇒

r 4 = 1 458 18 = 81 ⇒ r = ± 4 √81 ⇒ r = ±3

Adding terms in a geometric progression

The sum of the first n terms in a geometric progression is written as S n and is:

S n = a n · r – a1 r – 1 , r ≠ 1

8 In the geometric progression {6, 12, 24, 48, …}, find the sum of the first seven terms.

S n = a n · r – a1 r – 1 ⇒

S7 = a 7 · r – a1 r – 1 ; a1 = 6, r = 12 6 = 2

a n = a1 · r n – 1 ⇒ a 7 = 6 · 27 – 1 = 6 · 26 = 6 · 64 = 384

S7 = 384 · 2 – 6 2 – 1 = 762

384 × 2 6 = 762

Sum of infinite terms in a geometric progression with |r| < 1

The sum of all the terms in a geometric progression where |r | < 1 is:

S = a1 1 – r

In the geometric progression {3, 1, 1/3, ...}, the sum of the infinite terms is:

S = a1 1 – r ; a1 = 3, r = 1 3

S = 3 1 – 1/3 = 3 2/3 =

11. Find the general term for the geometric progressions:

a) 5, 15, 45, 135…

b) 6, 3, 3/2, 3/4…

12. For the geometric progression with first term a1 = 4 and the common ratio r = 5, find:

a) a6

b) a10

c) a n

13. Calculate the sum on the sides of infinite terms of the geometric progressions:

a) 1/5, 1/25, 1/125, 1/625…

b) 3, 2, 4/3, 8/9, 16/27…

14. In the geometric progression {2, 4, 8, 16, 32, …}, which term has the value of 1 024?

15. For the geometric progression where a4 = 135 and a6 = 1 215, find the common ratio.

16. Calculate the 10 first terms of the geometric progressions:

a) 2, 14, 98, 686…

b) 3, – 6, 12, –24…

17. The sum of infinite terms of a geometric progression is 6 and its first term is 4. Find the common ratio.

18. If the midpoints on the sides of an 8 m2 square are joined, another square is formed.

Calculate the areas of the series of smaller squares. What kind of sequence is this?

Energy-efficient heating

Carmen lives in Jaen with her parents, Antonio and Macarena, and her brother Manuel. This year, due to the economic situation, they must consider what type of heating they will use. To do this, the first step is to examine the most recent bills to compare the costs of the energy sources they can use in their home. They are as follows:

*Average price on latest bills

1 If they use 80 m³ of gas, with a conversion factor of 10.512 kWh/m³ and a VAT rate of 21%, how much would the gas bill be?

2 If a reduced VAT rate of 5% were applied, what would be the saving on the bill?

To work out the heating costs, we need to calculate the heat output for the home, which is the amount of heating that must be used to heat each of the rooms. To calculate this, we need to consider the factors that affect the temperature around the house: the surface area of the room (m²), the orientation of the house, its degree of insulation and its specific climate zone. Calculations should be made for each individual room, considering that the room height is no higher than 2.5 m. The following formula is used:

Heat output = S · O · A · Z · 85 W/m²

Where:

S: room area in square meters

O: coefficient dependent on the orientation

• North: 1.12

• South: 0.92

• East: 1

• West: 1

A: coefficient dependent on the insulation of the house:

• Good insulation: 0.93 (double glazing and double brick walls)

• Simple insulation: 1 (single glazing and double brick walls or double glazing and single brick walls)

• No insulation: 1.10 (single glazing and single brick walls)

Z: coefficient dependent on the climate zone. The following coefficients are determined by the CTE (Código Técnico de Edificación orTechnical Building Code)

• Zone A: 0.88

• Zone B: 0.95

• Zone C: 1.04

• Zone D: 1.12

• Zone E: 1.19

1 Work out the heating need for each room and the total for the house.

2 They have decided that, in principle, they are going to turn on the heating during the cool season in Jaén, which lasts approximately 3.7 months, from November 14th to March 5th. Work out the consumption in kWh for this period according to the heating needed, considering that the heating will be turned on for 8 hours per day.

Work out the consumption in kWh for this period according to the heating needed, considering that the heating will be turned on for 8 hours per day.

In Carmen´s house, in addition to the gas heating, they have air conditioning units in the living room and bedrooms. Carmen has done research on heat outputs. She explains to her parents that the heat output in a heat pump indicates its consumption in terms of the heat energy provided. In other words, it is a coefficient that correlates the energy used to the energy generated. It is obtained by dividing the heating output by its electrical consumption. For example, the heating appliances in Carmen’s house have a heat output coefficient of 4. This means that for every kW of energy used, they can provide 4 kW of heating energy.

The fixed costs of each system are shown in the table below.

Off-peak

€

kWh per day

2 Considering they have a contracted power of 4.6 kW, work out the heating cost if they were to use the heat pumps in the air-conditioning appliances. SECTION 3

1 Using the results from the previous exercises, work out the total heating if they used gas to heat the house.

2 What is the footprint of each family member? SECTION 4

If no green energy is contracted, the carbon footprint for the electricity consumption is 0.25 kg CO2 per kWh; and for gas consumption, the footprint is 2.03 kg CO2 per m³.

1 Work out the household’s carbon footprint for each energy source.

1 Prepare a budget for your household heating consumption.

2 Find more information on green energy and renewable energy. What is the difference?

1 Write the classification of real numbers and give three examples of each.

Solution

Real numbers (ℝ) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ Rational numbers (ℚ)

Irrational numbers: π, √ 2 , 3 √ 5 … ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Integers (ℤ)

Negative numbers: –1, –2,

Fractions: 2 3 , 7 6 , –3 2 … ⎧ ⎨ ⎩ Natural numbers ℕ: 0, 1, 2…

2 Work out the absolute and relative error when approximating the number π to 22/7. Round the result to four decimal places.

Solution

Absolute error: 0.0013 Relative error: 0.0004

3 Extract all possible factors from:

4 Four friends have enough food for a 15-day holiday. If two more friends join them, how many days’ worth of food will they have, considering they all eat the same amount each day? Solution

5 What is the interest rate on a €4 200 deposit that has accrued €196 in interest over a period of 14 months?

6 Work out the sum of the infinite terms in the following progression: 1/10, 1/100.... Solution

at = 1/10, r = 1/10 ⇒ s = 1/10 1 – 1/10 = 1/9

7 A square plot of land is sold for €3.6/m2. The total sale price was €3 802.5. What is the length of each side of the plot?

Solution 380 .5: 3.6 32.m 5 2 =

8 A square plot of land has an area of 756.25 m2. How much will it cost to fence the plot if each metre costs €5?

Solution

Length of side: L = 756 25 = 27.5 m ⇒ Perimeter = 4 · 27.5 = 110 m

Cost = 110 · 5 = €550

1. Define arithmetic progression and give an example.

2. Perform the following operations:

a) 3 5 ( 1 4 + 7 8 ) –5 4 b) ( 7 12 – 1) : 3 4 –5 3

3. Add and subtract the following radicals:

a) 332 – 2 50 + 72 b) 75 2 – 427 + 512

4. Ten workers can asphalt 80 km in 24 days. How many workers will it take to asphalt 220 km in 30 days?

5. Work out: We buy a TV set with a 20% discount and have to pay 21% VAT. If we paid €232 for the TV, what was the original price?

6. Find the sum of the first 25 terms of the progression whose general term is: a n = 4 n – 3

7. Aitana has covered 3/5 of the total distance of a route. She then covers 2/3 of the remaining distance and still has 6 km to go. What is the total length of the route?

8. The first three finishers in a competition must share €17 930. This sum must be divided into parts inversely proportional to their finishing position. How much will each competitor receive?

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