Inductive proof that every term is a sequence is divisible by 16 I have this question:

The $n$th member $a_n$ of a sequence is defined by $a_n = 5^n + 12n -1$. By considering $a_{k+1} - 5a_k$ prove that all terms of the sequence are divisible by 16. 

I can do the induction and have managed to rearrange the expression at the inductive step such that the expression must be divisible by 16. In other words, I can do the question fine. My question is: why must we consider $a_{k+1} - 5a_k$? Why can't we prove this by induction just by looking at $a_{k+1}$? Also, how can it be deduced that the expression we must consider is $a_{k+1} - 5a_k$?

A simple proof with some arithmetic modulo $16$:
$$\begin{array}{r|ccc}
n\equiv\pmod 4\quad & 5^n & 12n-1& a_n \\
\hline
0 &  1 & -1 & 0 \\
1 & 5 & 11 & 0 \\
2 & 9 & 7 & 0 \\
3 & 13 & 3 & 0
\end{array}$$
