Binomialllllllll (1 Viewer)

untouchablecuz · Mar 5, 2010

$\\$Sum the expression, $\\\binom{n}{0}+\binom{n}{4}+\binom{n}{8}+...+\binom{n}{n}\\$by considering the binomial expansions of $ (1+i)^n, (1-i)^n,(1+1)^n $ and $(1-1)^n $ (note that $n$ is a multiple of 4)$\\$

i got (by simply adding up all the expansions):

$\\\binom{n}{0}+\binom{n}{4}+\binom{n}{8}+...+\binom{n}{n}=\frac{1}{2}[\left2^{\frac{n}{2}}\cos\frac{n\pi}{4}+2^{n-1}]$

can someone do it and confirm (as i don't have answers), thanks

Pwnage101 · Mar 5, 2010

Just checked on a spreadsheet, took n as 10, 100 and 1000 and did the summation, and yes what you have stated seems to be correct.

Binomialllllllll (1 Viewer)

untouchablecuz

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