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TMUA 2018 · Paper 2 · Question 16 of 20

TMUA 2018 Paper 2 Question 16

Sequences and series — Medians of an arithmetic progression · parity of the common difference. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Sequences and seriesMedians of an arithmetic progression · parity of the common difference8 options
In this question, x1, x2, x3, … is an arithmetic progression, all of whose terms are integers.

Let n be a positive integer. If the median of the first n terms of the sequence is an integer, which of the following three statements must be true?

I    The median of the first n+2 terms is an integer.
II   The median of the first 2n terms is an integer.
III  The median of x2, x4, x6, …, x2n is an integer.

  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 · F
  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
For an arithmetic progression with first term a and common difference d, the median of the first n terms is a + (n−1)d2 whether n is odd or even. So when n is odd it is automatically an integer, and when n is even the hypothesis is telling us that d is even. Statement I is safe because n+2 has the same parity as n: either the median is automatic, or d is already known to be even. Statement III is also safe — those terms form an arithmetic progression with n terms, first term a+d and common difference 2d, whose median is a + nd. Statement II fails: take a = 0, d = 1, n = 1, where the median of the first term is 0 but the median of the first two is 12.