Question

prove that the constant term in the expansion of (x+1/x)^2n is 1.3.5.….(2n-1).2^n/n!

Answer 1

Hey there,

The 2 end terms would be a constant and x raised to the 2n power. Next to the constant term would be a term with x to the first power and next to the other end term would be x to the power of 2n-1.

So hopefully you see each term takes a step of 1 for the power of x. That power steps through the integers from 0 to 2n. That means the middle term will have x raised to the n power.

Using the binomial theorem, the nth power of x could be achieved (2nn)(2nn) ways. Since the constant and the x coefficient are both 1, that means we just take the number of ways times 1.

That means the middle term can be calculated like this:

(2nn)xn=(2n)!(n!)2xn(2nn)xn=(2n)!(n!)2xn

Hope this helps!

Answer 2

This is the step-by-step answer to this question. If you have any doubt, please feel free to ask. Good Luck! :)