If sum_{r=1}^{30} frac{r^2({}^{30}C_r)^2}{{}^ | vedclass

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

Question

(D) We have the expression $S = \sum_{r=1}^{30} \frac{r^2({}^{30}C_r)^2}{{}^{30}C_{r-1}}$.Using the property $\frac{{}^{n}C_r}{{}^{n}C_{r-1}} = \frac{

Vedclass · Accepted Answer

$465$

Vedclass · Answer

(D) We have the expression $S = \sum_{r=1}^{30} \frac{r^2({}^{30}C_r)^2}{{}^{30}C_{r-1}}$.Using the property $\frac{{}^{n}C_r}{{}^{n}C_{r-1}} = \frac{n-r+1}{r}$,we get $\frac{{}^{30}C_r}{{}^{30}C_{r-1}} = \frac{31-r}{r}$.Thus,the term becomes $r^2 \cdot {}^{30}C_r \cdot \frac{31-r}{r} = r(31-r) {}^{30}C_r$.Since $r \cdot {}^{30}C_r = 30 \cdot {}^{29}C_{r-1}$,the sum is $\sum_{r=1}^{30} 30 \cdot {}^{29}C_{r-1} (31-r)$.Let $k = r-1$,then $r = k+1$. As $r$ goes from $1$ to $30$,$k$ goes from $0$ to $29$.$S = 30 \sum_{k=0}^{29} {}^{29}C_k (30-k) = 30 \left( 30 \sum_{k=0}^{29} {}^{29}C_k - \sum_{k=0}^{29} k \cdot {}^{29}C_k \right)$.Using $\sum_{k=0}^{n} {}^{n}C_k = 2^n$ and $\sum_{k=0}^{n} k \cdot {}^{n}C_k = n \cdot 2^{n-1}$:$S = 30 \left( 30 \cdot 2^{29} - 29 \cdot 2^{28} \right) = 30 \cdot 2^{28} (60 - 29) = 30 \cdot 31 \cdot 2^{28} = 15 \cdot 31 \cdot 2^{29} = 465 \cdot 2^{29}$.Therefore,$\alpha = 465$.

If $\sum_{r=1}^{30} \frac{r^2({}^{30}C_r)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$,then $\alpha$ is equal to

Similar Questions

If $\sum_{k=1}^{10} k^{2} \binom{10}{k}^{2} = 22000 L$,then $L$ is equal to $.....$

The value of ${ }^{10} C_{1}+{ }^{10} C_{2}+{ }^{10} C_{3}+\ldots+{ }^{10} C_{9}$ is

$C_0 C_r + C_1 C_{r+1} + C_2 C_{r+2} + \dots + C_{n-r} C_n =$

The sum of the series $\frac{1}{1 \times 2} {}^{25}C_{0} + \frac{1}{2 \times 3} {}^{25}C_{1} + \frac{1}{3 \times 4} {}^{25}C_{2} + \ldots + \frac{1}{26 \times 27} {}^{25}C_{25}$ is

If $C_r = ^{100}C_r$,then the value of $1 \cdot C_0^2 - 2 \cdot C_1^2 + 3 \cdot C_2^2 - 4 \cdot C_3^2 + \dots + 101 \cdot C_{100}^2$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module