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TMUA 2016 · Paper 1 · Question 14 of 20

TMUA 2016 Paper 1 Question 14

Sequences and series — Sum of two geometric series · symmetric functions of the ratios. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Sequences and seriesSum of two geometric series · symmetric functions of the ratios6 optionshard
The terms of an infinite series S are formed by adding together the corresponding terms in two infinite geometric series, T and U.

The first term of T and the first term of U are each 4.

In order, the first three terms of the combined series S are 8, 3, and 54.

What is the sum to infinity of S?

  1. A325
  2. B203
  3. C645
  4. D403
  5. E16
  6. F32
Show the answer and worked solution
answer · D
  1. A325
  2. B203
  3. C645
  4. D403
  5. E16
  6. F32
Let the ratios be t and u. The second and third terms of S give 4t + 4u = 3 and 4t2 + 4u2 = 54, so t + u = 34 and t2 + u2 = 516. From (t+u)2 = t2 + u2 + 2tu we get 916 = 516 + 2tu, so tu = 18. There is no need to find t and u separately: S∞ = 41−t + 41−u = 4(2 − (t+u))1 − (t+u) + tu = 4 ⋅ 5438 = 403 (The ratios are in fact 12 and 14, so both series do converge.)