Question 13
The numbers in the sequence $101,$ $104,$ $109,$ $116,$ $\ldots$ are of the form $a_n=100+n^2,$ where $n=1,$ $2,$ $3,$ $\ldots.$ For each $n,$ let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}.$ Find the maximum value of $d_n$ as $n$ ranges through the positive integers.