if x={8^n - 7n - 1} and y={49(n-1)} prove that x is a subset of y

So presumably n is a positive integer.

You need to show that \(\displaystyle 8^n - 7n - 1 \equiv 0\ (\text{mod }49)\)

(The expression also must be non-negative but this is trivial.)

Use mathematical induction. Base case, n = 1.

\(\displaystyle 8^1 - 7(1) - 1 = 0 \equiv 0\ (\text{mod }49)\)

Induction step.

\(\displaystyle 8^{n+1} - 7(n+1) - 1 \equiv 8 \cdot 8^n - 7n - 7 - 1 \equiv (8)(7n+1) - 7n - 8 \equiv\)

\(\displaystyle 56n + 8 - 7n - 8 \equiv 49n \equiv 0\ (\text{mod }49)\)