If a_n = sum_{r=0}^n frac{1}{^nC_r},then sum_ | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Given $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$.Let $b_n = \sum_{r=0}^n \frac{r}{^nC_r}$.Using the property $^nC_r = ^nC_{n-r}$,we can write:$b_n = \fr

Vedclass · Accepted Answer

$\frac{1}{2}na_n$

Vedclass · Answer

(C) Given $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$.Let $b_n = \sum_{r=0}^n \frac{r}{^nC_r}$.Using the property $^nC_r = ^nC_{n-r}$,we can write:$b_n = \frac{0}{^nC_0} + \frac{1}{^nC_1} + \frac{2}{^nC_2} + \dots + \frac{n}{^nC_n}$.Also,$b_n = \frac{n}{^nC_n} + \frac{n-1}{^nC_{n-1}} + \dots + \frac{0}{^nC_0} = \sum_{r=0}^n \frac{n-r}{^nC_r}$.Adding both expressions for $b_n$:$2b_n = \sum_{r=0}^n \frac{r + (n-r)}{^nC_r} = \sum_{r=0}^n \frac{n}{^nC_r} = n \sum_{r=0}^n \frac{1}{^nC_r}$.Since $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$,we have $2b_n = na_n$.Therefore,$b_n = \frac{1}{2}na_n$.

If $a_n = \sum_{r=0}^n \frac{1}{^nC_r}$,then $\sum_{r=0}^n \frac{r}{^nC_r}$ equals

Similar Questions

The number of ways in which four letters of the word '$MATHEMATICS$' can be arranged is given by

$A$ man has $7$ relatives: $4$ ladies and $3$ gentlemen. His wife also has $7$ relatives: $3$ ladies and $4$ gentlemen. In how many ways can they invite a dinner party of $3$ ladies and $3$ gentlemen,such that there are $3$ of the man's relatives and $3$ of the wife's relatives?

The number of times the digit $3$ will be written when listing the integers from $1$ to $1000$ is

There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

On her vacations,Veena visits four cities $(A, B, C, D)$ in a random order. What is the probability that she visits $A$ just before $B$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module