Sum of powers is a perfect square

Question

Show that $ \sum_{k=1}^{2000} {2^k+7^k+9^k}$ îs a perfect square. I tried grouping terms or evaluating the geometric progressions...but without success. I got $$\frac{48∗2^{2000}+28∗7^{2000}+27∗9^{2000}−271}{24}$$with the formula of a geometric sum. How do i show this îs a square?

Answer 1

Look at the sums $S(q):=\sum_{k=1}^{2000}q^k$, $q=2,7,9$, modulo four.

With $q=2$ that sum is even but not a multiple of four, all due to the first term. With $q=9$ all the terms are congruent to $1\pmod4$. As $4\mid 2000$ we also have $4\mid S(9)$. With $q=7$ we see that the residues of the terms modulo four alternate between $+1$ and $-1$. Again implying that $4\mid S(7)$.

The conclusion is that $S(2)+S(7)+S(9)$ is even but not a multiple of four. Therefore it is NOT a square.