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|a_k| \leq \frac{k(n+1-k)}{2} when |a_{k-1} - 2a_{k} + a_{k+1}| \leq 1

Source: Singapore Senior Math Olympiad 2012 2nd Round p4 SMO

March 25, 2020

recurrence relationinequalitiesSequencealgebra

Problem Statement

Let

a_1, a_2, ..., a_n, a_{n+1}

be a finite sequence of real numbers satisfying

a_0 = a_{n+1} = 0

and

|a_{k-1} - 2a_{k} + a_{k+1}| \leq 1

for

k = 1, 2, ..., n

Prove that for

k=0, 1, ..., n+1,

|a_k| \leq \frac{k(n+1-k)}{2}