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TMUA 2020 · Paper 2 · Question 15 of 20

TMUA 2020 Paper 2 Question 15

Trigonometry — Summing a periodic sine series. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2TrigonometrySumming a periodic sine series6 options
Which one of the following is a necessary and sufficient condition for ∑k=1nsin(kπ3) = √32 to be true?
  1. An = 1
  2. Bn is a multiple of 3
  3. Cn is a multiple of 6
  4. Dn is 1 more than a multiple of 3
  5. En is 1 more than a multiple of 6
  6. Fn is 1 more than a multiple of 6 or n is 2 more than a multiple of 6
Show the answer and worked solution
answer · D
  1. An = 1
  2. Bn is a multiple of 3
  3. Cn is a multiple of 6
  4. Dn is 1 more than a multiple of 3
  5. En is 1 more than a multiple of 6
  6. Fn is 1 more than a multiple of 6 or n is 2 more than a multiple of 6
The terms repeat with period 6: for k = 1,  2,  3,  4,  5,  6 they are √32,  √32,  0,  −√32,  −√32,  0, and each block of six sums to zero. So work out the running totals across one block: S1 = √32, S2 = √3, S3 = √3, S4 = √32, S5 = 0, S6 = 0, and then the pattern repeats exactly. The value √32 occurs at n ≡ 1 and n ≡ 4 modulo 6, and those two residues together are precisely n ≡ 1 modulo 3. Stopping at n = 1 or at n ≡ 1 modulo 6 gives a sufficient but not necessary condition.