Question

Prove by induction 7^n+10 < 8^n

Answer 1

For n>=2

Answer 2

For n=2, 7^2 + 10 =59 < 64 so its true for n=2.

Assume 7^n + 10 <8^n for some n>=2.
Then

8(7^n+10) <8^(n+1)
(7+1)(7^n+10) < 8^(n+1)
7^(n+1)+ 7^n +70 + 10 < 8^(n+1)
7^(n+1) +10 < 8^(n+1) -7^n -70 < 8^(n+1)

So if the statement is true for n, its true for n+1, By induction, its true for all n>=2.