prove 3^n is greater or equal to 1+2^n

Question

principle of mathematical induction proofs

Answer 1

When n=0, 3ⁿ=2ⁿ=1, but the inequality is false because 3⁰<1+2⁰.

The inequality must be qualified by integer n>0.

When n=1, 3ⁿ=3 and 2ⁿ=2 so 3¹=1+2¹, and the inequality is true.

The base case is when n=1 (not n=0).

3ⁿ⁺¹=3.3ⁿ and 2ⁿ⁺¹=2.2ⁿ.

Assume 3ⁿ≥2ⁿ+1 then 3ⁿ⁺¹=3.3ⁿ≥3.2ⁿ+3.

3.2ⁿ=(1+2)2ⁿ=2ⁿ+2ⁿ⁺¹, 3ⁿ⁺¹≥2ⁿ+2ⁿ⁺¹+3.

Since 2ⁿ+3>1, 3ⁿ⁺¹≥2ⁿ⁺¹.