Hi. I am trying to prove the following:
1 + C(2n, 1) + C(2n, 2) + ... + C(2n, k) <= (2n)^k for n >= k > 1.
I would appreciate any suggestions you all may have.
Thanks.
You could prove it via Proof By Induction...
P(k)
$\displaystyle \displaystyle\ 1+\binom{2n}{1}+\binom{2n}{2}+....+\binom{2n}{k}\ \le\ (2n)^k\ ?$
P(k+1)
$\displaystyle \displaystyle\ 1+\binom{2n}{1}+\binom{2n}{2}+....+\binom{2n}{k}+\ binom{2n}{k+1}\ \le\ (2n)^{k+1}\ ?$
Try to prove that P(k+1) will be valid if P(k) is valid.
Proof
$\displaystyle \displaystyle\ (2n)^k+\binom{2n}{k+1}\ \le\ 2n(2n)^k\ ?$
$\displaystyle \displaystyle\binom{2n}{k+1}\ \le\ (2n-1)(2n)^k$
$\displaystyle \displaystyle\frac{[2n]!}{[2n-(k+1)]![k+1]!}\ \le\ (2n-1)(2n)(2n)(2n)....$
$\displaystyle \displaystyle\frac{(2n)(2n-1)(2n-2)....}{(k+1)!}\ \le\ (2n-1)(2n)(2n)(2n)....$
There are k+1 terms in the numerator of the LHS and k+1 terms on the RHS, since there are k occurances of 2n.
It's easy to see that the inequality holds.
The proof is then complete when you examine the base case for k=2, assuming k is an integer.
$\displaystyle \displaystyle\ 1+\binom{2n}{1}+\binom{2n}{2}\ \le\ (2n)^2\ ?$
$\displaystyle \displaystyle\ 1+2n+\frac{2n(2n-1)}{2}\ \le\ 4n^2\ ?$
$\displaystyle \displaystyle\ 1+2n+n(2n-1)\ \le\ 4n^2\ ?$
$\displaystyle 1+2n+2n^2-n\ \le\ 4n^2\ ?$
$\displaystyle 2n^2+n+1\ \le\ 4n^2\ ?$
$\displaystyle n+1\ \le\ 2n^2\ ?$
$\displaystyle n(2n-1)\ \ge\ 1\ ?$
$\displaystyle \text{\footnotesize\ True for \;\;}n\ \ge\ 1$