Finding the value of an expression involving co-efficients in binomial expansion

Question

Let $(1+x)^n = C_0 +C_1.x+C_2.x^2 +\ldots+C_n.x^n,$ $n$ being a positive integer. Then find the value of the following expression: $$\left(1+\frac{C_0}{C_1}\right)\left(1+\frac{C_1}{C_2}\right).....\left(1+\frac{C_{n-1}}{C_n}\right)$$

Answer 1

Hint: Look at a typical term $1+\dfrac{\binom{n}{i-1}}{\binom{n}{i}}$. This simplifies to $1+\dfrac{i}{n-i+1}$, and then to $\dfrac{n+1}{n-i+1}$. The rest is straightforward.

Remark: We have used the fact that $\binom{n}{k}=\frac{n!}{k!(n-)!}$. One can alternately use Pascal's Identity $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$. That takes somewhat more work.