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TMUA 2016 · Paper 1 · Question 4 of 20

TMUA 2016 Paper 1 Question 4

Sequences and series — Alternating sequence · simplifying before summing. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Sequences and seriesAlternating sequence · simplifying before summing7 options
The sequence an is defined by the rule an = (−1)n − (−1)n−1 + (−1)n+2 for n ≥ 1.

Find the value of ∑n=139 an

  1. A−39
  2. B−3
  3. C−1
  4. D0
  5. E1
  6. F3
  7. G39
Show the answer and worked solution
answer · B
  1. A−39
  2. B−3
  3. C−1
  4. D0
  5. E1
  6. F3
  7. G39
Do not list terms; simplify the rule first. Since (−1)n−1 = −(−1)n and (−1)n+2 = (−1)n, the three pieces become (−1)n + (−1)n + (−1)n, so an = 3(−1)n. The sequence is therefore −3,  3,  −3,  3,  …, and over 39 terms the pairs cancel leaving one extra −3 from the odd n. Formally ∑n=139(−1)n = −1, so the sum is 3 × (−1) = −3.