Prove, by induction, that $5^n + 5 < 5^{n+1}$ for all $n\in\Bbb N$. Prove, by induction, that $5^n + 5 &lt; 5^{n+1}$ for all $n\in\Bbb N$.

Attempt:

If $n = 1$, then $5^1 + 5 &lt; 5^{2}$ => $10 &lt; 25$ which is a true statement so the base case holds.

Assume $5^k + 5 &lt; 5^{k+1}$ is true for some $k\in\Bbb N$.

How to prove it for $n = k+1$?
$$5^{k+1} + 5 &lt; 5^{k+1} + 25 = 5(5^k + 5) &lt; 5 \cdot 5^{k+1} = 5^{k+2}$$
$$
\color{#0000ff}{\large 5^{n + 1} + 5} = 5\left(5^{n} + 5\right) - 20 &lt; 5\times 5^{n + 1} - 20 \color{#0000ff}{\large &lt; 5^{n + 2}}
$$