1. **State the problem:** Solve the equation $5^n = n + 2^n$ for $n$. 2. **Understand the equation:** We want to find the value(s) of $n$ such that the exponential expression $5^n$ equals the sum of $n$ and $2^n$. 3. **Check for integer solutions by testing small values:** - For $n=0$: $5^0=1$, $n+2^n=0+1=1$, so $1=1$ is true. - For $n=1$: $5^1=5$, $1+2^1=1+2=3$, so $5\neq3$. - For $n=2$: $5^2=25$, $2+2^2=2+4=6$, so $25\neq6$. 4. **Check behavior for larger $n$:** Since $5^n$ grows faster than $2^n$ and $n$, for $n>2$, $5^n$ will be much larger than $n+2^n$, so no equality. 5. **Check for negative $n$:** - For $n=-1$: $5^{-1}=\frac{1}{5}=0.2$, $-1+2^{-1}=-1+0.5=-0.5$, so $0.2\neq-0.5$. 6. **Conclusion:** The only solution is $n=0$. **Final answer:** $$n=0$$