13. $[8]$ Find the smallest positive integer $n$ such that $\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$.
Answer: 7 Writing $5^{n+1}=5 \cdot 5^{n}$ and $2^{n+1}=2 \cdot 2^{n}$ and cross-multiplying yields $0.01 \cdot 5^{n}>2.99 \cdot 2^{n}$, and re-arranging yields $(2.5)^{n}>299$. A straightforward calculation shows that the smallest $n$ for which this is true is $n=7$.