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TMUA 2022 · Paper 1 · Question 8 of 20

TMUA 2022 Paper 1 Question 8

Sequences and series — Geometric series · relating partial sums. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 1Sequences and seriesGeometric series · relating partial sums5 options
A geometric sequence has first term a and common ratio r, where a and r are positive integers and r is greater than 1.

The sum of the first n terms of this sequence is denoted by Sn.

It is given that the terms of the sequence satisfy S30 − S20 = k S10 for some positive integer k.

What is the smallest possible value of k?

  1. A210
  2. B220
  3. C230
  4. D210210−1
  5. E210(210−1)
Show the answer and worked solution
answer · B
  1. A210
  2. B220
  3. C230
  4. D210210−1
  5. E210(210−1)
With Sn = a(rn − 1)r−1, the difference S30 − S20 = a(r30 − r20)r−1 = r20⋅a(r10−1)r−1 = r20S10. So k = r20 exactly, and since r is an integer greater than 1 the smallest case is r = 2, giving k = 220.