Induction Proof: $5^n + 5 < 5^{n+1}$ I am trying to prove for all natural $n$ that:
$$5^n + 5 < 5^{n+1}$$
I did the basic step with $n=1$ and inequality holds, I am now at the induction step:
$$5^{k+1} + 5 < 5^{k+2}$$
and I have no idea how to proceed from here. Can someone give me a clue?

Inequality holds for $n=1$.
For $n=k$
$$5^k + 5 < 5^{k+1}$$
We can write this as
$$5^{k+1} - (5^k + 5) = p$$
where $p$ is a positive integer.
$$5*5^k - 5^k - 5 = p$$
$$4*5^k - 5 = p$$
$$5^k=\frac{p+5}{4}$$
Now we have to prove it is true for $n = k+1$. So if $(5^{k+2} - (5^{k+1}+5))$ is positive then inequality holds true for $n=k+1$.
$$5^{k+2} - (5^{k+1}+5)$$
$$25*5^k - (5*5^{k}+5)$$
$$25*5^k - 5*5^{k}-5$$
$$20*5^k - 5$$
substitute value of $5^k$,
$$20*(\frac{p+5}{4}) - 5$$
$$5*(p+5)-5$$
$$5p+25-5$$
$$5p+20$$
$$5(p+4)$$
As $p$ is a positive integer, so $5(p+4)$ is also positive integer.
Thus, we can say that 
$$5^{k+1}+5 < 5^{k+2}$$
