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TMUA 2022 · Paper 2 · Question 8 of 20

TMUA 2022 Paper 2 Question 8

Sequences and series — Arithmetic sequences · guaranteeing a pair. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Sequences and seriesArithmetic sequences · guaranteeing a pair6 optionshard
A selection, S, of n terms is taken from the arithmetic sequence 1,  4,  7,  10,  …,  70.

Consider the following statement:

(×) There are two distinct terms in S whose sum is 74.

What is the smallest value of n for which (×) is necessarily true?

  1. A12
  2. B13
  3. C14
  4. D21
  5. E22
  6. F23
Show the answer and worked solution
answer · C
  1. A12
  2. B13
  3. C14
  4. D21
  5. E22
  6. F23
The sequence is 1 + 3k for k = 0 to 23, so it has 24 terms. Two terms 1+3k and 1+3m sum to 74 exactly when k + m = 24, and with k ≠ m that gives the eleven pairs k = 1,  …,  11 matched with m = 23,  …,  13. The terms with k = 0 and k = 12 belong to no pair. A selection avoiding every pair can take both loners and one from each of the eleven pairs — thirteen terms. So thirteen is not enough and n = 14 forces a pair.