Problem 5

Consider an infinite sequence

a_1,a_2,\ldots

of positive integers such that

100!(a_m+a_{m+1}+\cdots+a_n)

is a multiple of

a_{n-m+1}a_{n+m}

for all positive integers

m,n

with

m\le n

. Prove that the sequence is either bounded or linear. A sequence is bounded if there is a constant

N

such that

a_n<N

for every positive integer

n

, and it is linear if

a_n=n\cdot a_1

for every positive integer

n

Step 6 of 10: The preceding terms are unbounded

a_{kd}=kt\Longrightarrow a_{kd-1}\text{ is unbounded}

Detailed analysis

From Step 5, $a_{kd}=kt$ tends to infinity. But $a_{kd}=a_{1+(kd-1)}$ divides $c(a_1+a_{kd-1})$ by Step 1, so $a_{kd-1}$ is unbounded as $k$ varies.