Prove that n is less than 2^n by induction?

Answer 1

check for 1:
1 < 2

assume it is true for n:=k
k < 2^k
prove it is true for k+1

k + 1 < 2^(k+1)
k + 1 < 2^k + 2

we know that k < 2k and 1 < 2 therefore LHS < RHS

Good Luck

Answer 2

If n =1, then 2^n = 2, so that n < 2^n.

Suppose the inequality is true for some positive integer n. Then, by the inductive assumption,

2^(n+1) = 2 * 2^n > 2n. Since n >=1, 2n = n + n >= n +1, so that
2^(n+1) > 2n >= n +1 ==> n +1 < 2^(n +1), proving the inequality holds for the integer n +1.

This completes the induction and shows n < 2^n for every positive integer n.