240. Geometric progression. One morning, Professor Rackbrain challenged his friends to find no fewer than three integers that form a geometric progression starting with 1, and the sum of which must be a perfect square. (For example, $1+2+4+8+16+32=63$. However, the last number is one short of being a square. I know only two solutions in integers; both are quite easy to find.)
240.

\[
1+3+9+27+81=121=11^{2}
\]

and

\[
1+7+49+343=400=20^{2}
\]