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Sum of a geometric series, finite and infinite sheet
5-mark answer frame

Find the sum to n terms of a series like 7 + 77 + 777 + ....

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found. Write the series and 'To find: Sₙ'.
  2. 2 · Write the formula or theorem. For a GP, Sₙ = a(rⁿ − 1)/(r − 1), r ≠ 1.
  3. 3 · Take out the repeated digit and make nines. Sₙ = 7(1 + 11 + 111 + ...) = (7/9)(9 + 99 + 999 + ...).
  4. 4 · Split into a GP. (7/9)[(10 + 10² + ... + 10ⁿ) − n].
  5. 5 · Use the GP sum. 10 + 10² + ... + 10ⁿ = 10(10ⁿ − 1)/9.
  6. 6 · Simplify. Sₙ = 70(10ⁿ − 1)/81 − 7n/9.
  7. 7 · Check and box the answer. Put n = 1 or n = 2 and compare with 7 or 7 + 77 = 84.

Scoring tip: Do not forget the '− n': one 1 is subtracted from each of the n brackets.