1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of (nCo/n)+(nC1/n+1)+(nC2/n+2)+. . .+(nCn/2n) is equal to

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The value of $\frac{^{n}C_{o}}{n}+\frac{^{n}C_{1}}{n+1}+\frac{^{n}C_{2}}{n+2}+. . .+\frac{^{n}C_{n}}{2n} $ is equal to

Binomial Theorem

Solution:

$S=\frac{^{n}C_{0}}{n}+\frac{^{n}C_1}{n+1}+\frac{^{n}C_{2}}{n+2}+...+\frac{^{n}C_{n}}{2n}$
$=\left(\frac{1}{x}\right)^{r}=\,{}^{m}C_{r} x^{2m-3r} \,{}^{n}C_{1} \int\limits_{0}^{1} x\,{}^{n} dx + ... + \,{}^{n}C_{n} \int\limits_{0}^{1} x^{2n-1} dx$
$ =\int\limits_{0}^{1} \left[^{n}C_{0}x \,{}^{n-1}+ \,{}^{n}C_{1}x^{n}+...+ \,{}^{n}C_{n}x^{2n-1}\right]dx$
$ =\left(\frac{1}{x}\right)^{r}=\,{}^{m}C_{r}x^{2m-3r}$
$=\int\limits_{1}^{2} x^{n}\left(x-1\right)^{n-1}dx$