Question

Simplify:

$\left(\frac{c_0+c_1}{c_0}\right)\left(\frac{c_1+c_2}{c_1}\right)\left(\frac{c_2+c_3}{c_2}\right).....\left(\frac{c_{n-1}+c_n}{c_{n-1}}\right)$

Answer 1

We have,

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

$\Rightarrow (1+\frac{C_1}{C_0})(1+\frac{C_2}{C_1})(1+\frac{C_3}{C_2})..........(1+\frac{C_n}{C_{n-1}})$

$\Rightarrow (1+\frac{n}{1})(1+\frac{n(n-1)}{n2!})..................(1+\frac{1}{n})$

$\Rightarrow \left(\frac{1+n}{1}\right)\left(\frac{1+n}{1}\right)\left(\frac{1+n}{1}\right).................\left(\frac{1+n}{1}\right)$

$\Rightarrow \frac{{(1+n)}^n}{n!}$

Hence, this is the answer.