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Q.
The statement $P(n) = 9^n - 8^n$ , when divided 8, always leaves the remainder
Principle of Mathematical Induction
Solution:
$P(n)=9^n -8^n$
$P(1) = 9 - 8 = 1$
$P ( 1 )-1 = 0$ which is divisible by 8
$\therefore \, P(1) = 1$ is the remainder, when $P(n)$ is divided by 8.
$P(2) = 9^2 - 8^2$
= 17 = 16 + 1
$\rightarrow$ remainder is 1, when divided by 8.