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TMUA 2018 · Paper 1 · Question 2 of 20

TMUA 2018 Paper 1 Question 2

Sequences and series — Arithmetic series · equal partial sums. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1Sequences and seriesArithmetic series · equal partial sums6 options
An arithmetic progression has first term a and common difference d.

The sum of the first 5 terms is equal to the sum of the first 8 terms.

Which one of the following expresses the relationship between a and d?

  1. Aa = −383d
  2. Ba = −7d
  3. Ca = −6d
  4. Da = 6d
  5. Ea = 7d
  6. Fa = 383d
Show the answer and worked solution
answer · C
  1. Aa = −383d
  2. Ba = −7d
  3. Ca = −6d
  4. Da = 6d
  5. Ea = 7d
  6. Fa = 383d
If S5 = S8 then the terms in between contribute nothing: u6 + u7 + u8 = 0. Those terms are a + 5d, a + 6d and a + 7d, so 3a + 18d = 0 and a = −6d. Using the sum formula directly gives the same thing: 52(2a + 4d) = 82(2a + 7d) reduces to 6a + 36d = 0.