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TMUA 2020 · Paper 1 · Question 4 of 20

TMUA 2020 Paper 1 Question 4

Sequences and series — GP and AP · terms shared between two progressions. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Sequences and seriesGP and AP · terms shared between two progressions6 options
The 1st, 2nd and 3rd terms of a geometric progression are also the 1st, 4th and 6th terms, respectively, of an arithmetic progression.

The sum to infinity of the geometric progression is 12.

Find the 1st term of the geometric progression.

  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
Show the answer and worked solution
answer · D
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
  6. F6
Let the GP be a,  ar,  ar2 and the AP have first term a and common difference d. Matching positions, a + 3d = ar and a + 5d = ar2, so 3d = a(r−1) and 5d = a(r2−1) = a(r−1)(r+1). Dividing (the sum to infinity exists, so r ≠ 1) gives 53 = r + 1, hence r = 23. Then a1 − 23 = 3a = 12, so a = 4.