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TMUA 2021 · Paper 1 · Question 3 of 20

TMUA 2021 Paper 1 Question 3

Sequences and series — Arithmetic and geometric progressions · simultaneous conditions. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 1Sequences and seriesArithmetic and geometric progressions · simultaneous conditions8 options
An arithmetic progression and a convergent geometric progression each have first term 12

The sum of the second terms of the two progressions is 12

The sum of the third terms of the two progressions is 18

What is the sum to infinity of the geometric progression?

  1. A−2
  2. B−1
  3. C−12
  4. D−13
  5. E13
  6. F12
  7. G1
  8. H2
Show the answer and worked solution
answer · G
  1. A−2
  2. B−1
  3. C−12
  4. D−13
  5. E13
  6. F12
  7. G1
  8. H2
Write the arithmetic progression as 12,  12 + d,  12 + 2d and the geometric one as 12,  12 r,  12 r2. The second-term condition is (12 + d) + r2 = 12, so d = −r2. The third-term condition gives 12 + 2d + r22 = 18; substituting d and multiplying by 8 gives 4r2 − 8r + 3 = 0, so r = 12 or r = 32. Convergence rules out 32, so r = 12 and the sum to infinity is 1/21 − 1/2 = 1.