Solve the following?

Let (1+x)^n=C_0+C_1x+C_2x^2+......+C_nx^n

Then the value of (1+C_0/C_1)*(1+C_1/C_2).....*(1+C_(n-1)/C_n) is?

1 Answer
Apr 15, 2018

(n+1)^n/(n!)

Explanation:

(1+C_0/C_1)(1+C_1/C_2)...(1+C_(n-1)/C_n)

The binomial theorem says that the k^(th) coefficient in the n^(th) order expansion of (1+x) is

C_k=(n!)/(k!(n-k)!)

This means that

C_k/C_(k+1)=((k+1)!(n-(k+1))!)/(k!(n-k)!)

=((k+1)!(n-k-1)!)/(k!(n-k)!)=(k+1)/(n-k)

So

1+C_k/C_(k+1)=1+(k+1)/(n-k)=(n-k+k+1)/(n-k)=(n+1)/(n-k)

So our product looks like

((n+1)/(n-0))((n+1)/(n-1))((n+1)/(n-2))...((n+1)/(1))=(n+1)^n/(n!)