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TMUA 2017 · Paper 1 · Question 11 of 20

TMUA 2017 Paper 1 Question 11

Sequences and series — Recurrence relations · periodic sequences. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Sequences and seriesRecurrence relations · periodic sequences5 options
The sequence xn is defined by the rules x1 = 7 xn+1 = 23xn − 535xn + 1 The first three terms in the sequence are 7,  3,  1

What is the value of x100?

  1. A−5
  2. B0
  3. C1
  4. D3
  5. E7
Show the answer and worked solution
answer · A
  1. A−5
  2. B0
  3. C1
  4. D3
  5. E7
Carry the recurrence on until it repeats. From x3 = 1, x4 = 23 − 535 + 1 = −5, and then x5 = −115 − 53−25 + 1 = −168−24 = 7 = x1. So the sequence is periodic with period 4: 7,  3,  1,  −5,  7,  3,  …. Since 100 is a multiple of 4, x100 = x4 = −5.