12/13/2017
OCR Further Maths Additional Pure Book
u k + 2 = 3u k + 1 − 2u k = 3(2 k + 1 − 1) − 2(2 k − 1) = 3 × 2k + 1 − 2 × 2k − 1 = 3 × 21 × 2k − 2 × 2k − 1 = 4 × 2k − 1 = 22 × 2k − 1 = 2k + 2 − 1 hence true for all positive integers.
E
Hence it is true for n = k + 1 and so it is true for all positive integers n.
Rewind
PL
The method of induction was covered in Pure Core Student Book 1, Chapter 6.
WORKED EXAMPLE 1.3
Consider the sequences u n =
1
1 . and t = n n 2n − 1
SA M
a Write down the first five terms of each sequence.
b State the value to which each sequence converges. c Find the sum of the first five terms and the sum to infinity of each sequence. d Describe each sequence.
1 1 1 1 1 a 1 1 1 1 1 , , , , and , , , , . 1 2 4 8 16 1 2 3 4 5
Substituting n = 1, 2, 3, …
b They both converge to 0.
As n → ∞,
c u n is a geometric sequence:
( ()) 1
1× 1−
S5 =
a(1 −
r n)
1−r
=
1−
r=
5
2
() 1
=
1 ∞
→0
1 2
31
16
2
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