TTMUA Lab
TMUA 2018 · Paper 1 · Question 8 of 20

TMUA 2018 Paper 1 Question 8

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

TMUA 2018 · Paper 1Sequences and seriesGeometric series · sums of squares and cubes6 options
The sum to infinity of a geometric progression is 6.

The sum to infinity of the squares of each term in the progression is 12.

Find the sum to infinity of the cubes of each term in the progression.

  1. A8
  2. B18
  3. C24
  4. D2167
  5. E72
  6. F216
Show the answer and worked solution
answer · D
  1. A8
  2. B18
  3. C24
  4. D2167
  5. E72
  6. F216
Squaring the terms of a geometric progression with first term a and ratio r gives another geometric progression, with first term a2 and ratio r2. So a1−r = 6 and a21−r2 = 12. Dividing the second by the square of the first, 1−r1+r = 1236 = 13, so 3 − 3r = 1 + r and r = 12, giving a = 3. The cubes form a geometric progression with first term 27 and ratio 18, summing to 271 − 18 = 2167.