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TMUA 2019 · Paper 2 · Question 11 of 20

TMUA 2019 Paper 2 Question 11

Sequences and series — Arithmetic series · what a fixed sum forces. Try it first; the answer and a full worked solution are below.

TMUA 2019 · Paper 2Sequences and seriesArithmetic series · what a fixed sum forces8 options
An arithmetic series has n terms, all of which are integers.

The sum of the series is 20.

Which of the following statements must be true?

I    The first term of the series is even.
II   n is even.
III  The common difference is even.

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · A
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
One well-chosen example can kill all three at once. Take the eight consecutive integers −1,  0,  1,  2,  3,  4,  5,  6: they are in arithmetic progression and sum to 20, yet the first term −1 is odd and the common difference 1 is odd, so I and III both fail. For II, take the five consecutive integers 2,  3,  4,  5,  6, which also sum to 20 with n = 5 odd. Nothing here is forced, so the answer is none of them.