The Hampshire College Summer Studies in Mathematics program has what it calls an “Interesting Test” that interested students are invited to take as part of their application. Their web page offers a sample of past test problems. One of these, under the heading “Problems which at first seem to lack sufficient information”, is as follows:

From the third term on, each term of the sequence of real numbers $latex a_1, a_2, ldots, a_{10}$ is the sum of the preceding two terms; that is, $latex a_n = a_{n-1} + a_{n-2}$ for $latex n = 3, 4, 5, ldots, 10$. If $latex a_7 = 17$, what is the sum of all the ten terms?

I’d heard this one before (though it’s hard to find a source for things like this – the cloesst I could find was this Math StackExchange question. The solution is as follows: we can write $latex…