Question

use mathematical induction
2^n>n^3

Answer 1

counterexample:
n=2
2^2=4
2^3=8
4 is not greater than 8

Answer 2

probably missing that a condition that n>= 10

Answer 3

oh:: 2^n>n^3
for every integer n>=10

Answer 4

\[(n+1)^3=n^3+3n^2+3n+1\]

\[3n^2+3n+1\leq 3n^2+3n^2+n^2=7n^2\leq n^3\]

Answer 5

wat is leq

Answer 6

i would have done:
\[n=10: 2^{10}>10^3\]
now assume for some k>10 the following holds: \[n=k: 2^k>k^3\]
now we want to show it is true k+1
\[n=k+1: 2^{k+1}=2^k2=2\cdot2^k>2 \cdot k^3>k^3\]

Answer 7

don't you need \[2^{k+1}>(k+1)^3\]