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TMUA 2017 · Paper 2 · Question 3 of 20

TMUA 2017 Paper 2 Question 3

Sequences and series — Geometric series · sum to infinity with surds. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Sequences and seriesGeometric series · sum to infinity with surds7 options
The first term of a geometric progression is 2√3 and the fourth term is 94.

What is the sum to infinity of this geometric progression?

  1. A−2(2−√3)
  2. B4(2√3 − 3)
  3. C16(8√3 + 9)37
  4. D4(2√3−3)7
  5. E4(2√3+3)7
  6. F2(2+√3)
  7. G4(2√3+3)
Show the answer and worked solution
answer · G
  1. A−2(2−√3)
  2. B4(2√3 − 3)
  3. C16(8√3 + 9)37
  4. D4(2√3−3)7
  5. E4(2√3+3)7
  6. F2(2+√3)
  7. G4(2√3+3)
The fourth term is ar3, so r3 = 9/42√3 = 98√3 = 3√38. Recognise 3√3 = (√3)3, so r3 = (√32)3 and r = √32, which is less than 1 so the series converges. Then S∞ = 2√31 − √32 = 4√32−√3, and multiplying top and bottom by 2+√3 gives 4√3(2+√3) = 8√3 + 12 = 4(2√3+3).