Problem 5

Determine all sequences

a_0,a_1,a_2,\ldots

of positive integers with

a_0\ge2015

such that for all integers

n\ge1

: (i)

a_{n+2}

is divisible by

a_n

; (ii)

|s_{n+1}-(n+1)a_n|=1

, where

s_{n+1}=a_{n+1}-a_n+a_{n-1}-\cdots+(-1)^{n+1}a_0

Step 5 of 5: Identify and verify both families

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a_n=c(n+2)n!\quad(n\ge1),\qquad a_0=c+1\text{ or }c-1

Detailed analysis

The ratio at n=1 makes a_1 a multiple of three; write a_1=3c and induct to obtain the displayed formula. The condition at n=1 gives a_0=c plus or minus one. The lower bound on a_0 gives c at least 2014 in the plus case and at least 2016 in the minus case. Finally, using the factorial identity for the alternating sum verifies both families.