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TMUA 2023 · Paper 1 · Question 4 of 20

TMUA 2023 Paper 1 Question 4

Sequences and series — Sum to infinity · trigonometric terms. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 1Sequences and seriesSum to infinity · trigonometric terms5 options
Evaluate ∑n=0∞ sin(nπ + π3)2n
  1. A0
  2. B13
  3. C√33
  4. D√3
  5. E3
Show the answer and worked solution
answer · C
  1. A0
  2. B13
  3. C√33
  4. D√3
  5. E3
Adding a whole number of half-turns only flips the sign: sin(nπ + θ) = (−1)n sinθ. With θ = π3 every term carries sinπ3 = √32, so the sum is √32∑n=0∞(−12)n. That is geometric with ratio −12, summing to 11+12 = 23. The total is √32⋅23 = √33.