Problem 3

For each integer

k\ge2

, determine all infinite sequences of positive integers

a_1,a_2,\ldots

for which there exists a polynomial

P

of the form

P(x)=x^k+c_{k-1}x^{k-1}+\cdots+c_1x+c_0

, where

c_0,c_1,\ldots,c_{k-1}

are non-negative integers, such that

P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k}

for every integer

n\ge1

Step 2 of 6: A telescoping identity

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In plain words

Writing the defining relation for two consecutive indices and dividing cancels most of the product, leaving a clean relationship between $a_n$ and $a_{n+k}$ .

\dfrac{P(a_n)}{P(a_{n-1})}=\dfrac{a_{n+k}}{a_n}

Detailed analysis

The relation gives $P(a_{n-1})=a_na_{n+1}\cdots a_{n+k-1}$ and $P(a_n)=a_{n+1}a_{n+2}\cdots a_{n+k}$ . Dividing, all the shared factors $a_{n+1},\ldots,a_{n+k-1}$ cancel, leaving $P(a_n)/P(a_{n-1})=a_{n+k}/a_n$ .