Question

Prove by PMI that
3^n >= 1+2^n

Answer 1

Hello,


`If\ n=1\ then\ 3^1 \geq 1+2^1\ since\ 3 \geq 3\\ \textrm{This proves the base case.}\\ \textrm{Now assume this holds for some n = k.}\\ \textrm{We need to show this holds for n = k +1}\\ 3^k \geq 1+2^k\ is\ true.\\ 3^(k+1)=3*3^k \geq 3*(1+2^k)\\ 3^(k+1)=3+(1+2)*2^k=3+2^k+2^(k+1) \geq 3+2^(k+1)\\ 3^(k+1) \geq 1+2^(k+1)\ is\ true.`