Hi everyone, I'm running into a snag on this problem and am asking for assistance. Sorry that this is a jpg, I have no idea how to use the Math Formula options.
While it does appear that this is what I'm looking for:
$\displaystyle
n = (a_n+1)^n \geq 1 + na_n + \tfrac{1}{2}n(n-1)a_n^2 \geq \tfrac{1}{2}n(n-1)a_n^2
$
Can you tell me how you got the middle equation? The problem I'm having with this current book is that it does exactly what you did here. It shows me a resolution without showing any of the scratch.
(Here $\displaystyle n\geq 2$).
This follows by binomial theorem: $\displaystyle (a+b)^n = \sum_{k=0}^n {n\choose k}a^kb^{n-k}$
If $\displaystyle a,b\geq 0$ then $\displaystyle (a+b)^n = \sum_{k=0}^2 {n\choose k} a^k b^{n-k} + \sum_{k=2}^n {n\choose k}a^kb^{n-k} \geq \sum_{k=0}^2 {n\choose k}a^k b^{n-k} =$$\displaystyle a^n + nab^{n-2}+\tfrac{1}{2}n(n-1)a^2b^{n-2}$.