12.5.3 Olympic algebra

Problem: Simplify the sum: \[ 1-\frac{1}{5}+\frac{1}{4}-\frac{1}{25}+\frac{1}{16}-\frac{1}{125}+\frac{1}{64}-\cdots-\frac{1}{5^{n}}+\frac{1}{4^{n}} . \] Ellipses and summation signs can't be used in the answer.