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TMUA 2022 · Paper 2 · Question 16 of 20

TMUA 2022 Paper 2 Question 16

Inequalities and reasoning — Inequalities between sequences. Try it first; the answer and a full worked solution are below.

TMUA 2022 · Paper 2Inequalities and reasoningInequalities between sequences8 options
In this question, a1,  …,  a100 and b1,  …,  b100 and c1,  …,  c100 are three sequences of integers such that an ≤ bn + cn for each n.

Which of the following statements must be true?

I   (minimum of  a1, …, a100) ≤ (minimum of  b1, …, b100) + (minimum of  c1, …, c100)
II  (minimum of  a1, …, a100) ≥ (minimum of  b1, …, b100) + (minimum of  c1, …, c100)
III (maximum of  a1, …, a100) ≤ (maximum of  b1, …, b100) + (maximum of  c1, …, c100)

  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 · D
  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
Only III survives. If a attains its maximum at index i, then ai ≤ bi + ci ≤ max b + max c. For I, take b = (0,  10,  …) and c = (10,  0,  …) with a equal to 10 throughout: both minima are 0, but min a = 10. For II, take b and c constantly 5 and a constantly 0: then min a = 0 but the right-hand side is 10.