Prove using the principle of mathematical induction that
3^n > 4n^2 +5n for n=4,5...
thank you!
This is very straightforward.
$\displaystyle 3^n>4n^2+5n$ ?
If this is true, we try to use it to prove if
$\displaystyle 3^{n+1}>4(n+1)^2+5(n+1)$
$\displaystyle (3)3^n>4(n^2+2n+1)+5(n+1)$ ?
$\displaystyle 3^n+(2)3^n>4n^2+4(2n)+4+5n+5$ ?
$\displaystyle 3^n+[(2)3^n]>4n^2+5n+[4(2n)+9]$
Therefore, we only need ask if
$\displaystyle 2(3^n)>4(2n)+9,\ n\ \ge\ 4$
$\displaystyle 3^n+3^n>4n+4.5+4n+4.5$
As $\displaystyle n^2>n$ and $\displaystyle 5n>4.5$, for n=4 and above,
only the first term needs to be tested.