Question

Evaluate :
$\left(\frac{C_0+C_1}{C_0}\right)\left(\frac{C_1+C_2}{C_1}\right)\left(\frac{C_2+C_3}{C_2}\right)\left(\frac{C_3+C_4}{C_3}\right)........\left(\frac{C_{n-1}+C_n}{C_{n-1}}\right)$

Answer 1

We know, for any $k<n$ ,$\frac{C_{k+1}}{C_k}$

$=\frac{n!\left(n-k\right)!k!}{\left(k+1\right)!\left(n-k-1\right)!n!}$

$=\frac{\left(n-k\right)\left(n-k-1\right)!k!}{\left(k+1\right)k!\left(n-k-1\right)!}=\frac{n-k}{k+1}$

Now, $\frac{C_k+C_{k+1}}{C_k}=1+\frac{C_{k+1}}{C_k}=1+\frac{n-k}{k+1}=\frac{n-k+k+1}{k+1}=\frac{n+1}{k+1}$

Then, $\left(\frac{C_0+C_1}{C_0}\right)\left(\frac{C_1+C_2}{C_1}\right).....\left(\frac{C_n+C_n}{C_{n-1}}\right)$

$=\frac{n+1}{0+1}.\frac{n+1}{1+1}.....\frac{n+1}{\left(n-1\right)+1}=\frac{{\left(n+1\right)}^n}{\mathrm{1.2......}n}=\frac{{\left(n+1\right)}^n}{n!}$

$\left(\frac{C_0+C_1}{C_0}\right)\left(\frac{C_1+C_2}{C_1}\right)....\left(\frac{C_{n-1}+C_n}{C_{n-1}}\right)=\frac{{\left(n+1\right)}^n}{n!}$