Hi,

I'm having trouble with this problem:

Use mathematical induction to show that:

$\displaystyle R_{n} : 5^n \geq n^5$ for all $\displaystyle n \geq 5.$

It suggests using this inequality as the basis for an approximation:

$\displaystyle (n+1)^5 = n^5 + 5n^4 + 10n^3 + (10n^2 + 5n + 1) < n^5 + n^5 + 2n^5 + n^5$

This is what I've done so far:

$\displaystyle (n+1)^5 = n^5 + 5n^4 + 10n^3 + (10n^2 + 5n + 1) < n^5 + n^5 + 2n^5 + n^5$

is equal to

$\displaystyle (n+1)^5 < 5n^5$

If

$\displaystyle 5^k \geq k^5$

then

$\displaystyle 5(5)^k \geq 5k^5$

which is equal to

$\displaystyle 5^{(k+1)} \geq 5k^5$

Since

$\displaystyle (n+1)^5 < 5n^5$

$\displaystyle 5^{(k+1)} \geq 5k^5 > (n+1)^5$

Therefore

$\displaystyle 5^{(k+1)} > (n+1)^5$

My problem is that I don't know how to get from

$\displaystyle 5^{(k+1)} > (n+1)^5$

to

$\displaystyle 5^{(k+1)} \geq (n+1)^5$