(D) The given series is $S = \sum_{r=0}^{n} (a + rb)C_r$.This can be split into two parts:$S = a \sum_{r=0}^{n} C_r + b \sum_{r=0}^{n} r C_r$.We know

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(D) The given series is $S = \sum_{r=0}^{n} (a + rb)C_r$.This can be split into two parts:$S = a \sum_{r=0}^{n} C_r + b \sum_{r=0}^{n} r C_r$.We know that $\sum_{r=0}^{n} C_r = 2^n$ and $\sum_{r=0}^{n} r C_r = n 2^{n-1}$.Substituting these values:$S = a(2^n) + b(n 2^{n-1})$.Factoring out $2^{n-1}$:$S = 2a(2^{n-1}) + bn(2^{n-1})$.$S = (2a + nb) 2^{n-1}$.

Class 11 Mathematics Binomial Theorem Properties of binomial coefficients

The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + \dots + (a + nb)C_n$ is,where $C_r$ denotes the combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$.

A
$(a + 2nb)2^n$
B
$(2a + nb)2^n$
C
$(a + nb)2^{n - 1}$
D
$(2a + nb)2^{n - 1}$

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Class 11 Questions

Class 11

Mathematics Questions

Chapter

Binomial Theorem

Subtopic

Properties of binomial coefficients

$3 \cdot C_0 + 7 \cdot C_1 + 11 \cdot C_2 + \ldots + (3 + 4n) C_n =$

MediumAP EAMCET 2017

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If $\sum\limits_{K = 1}^{12} {12K \cdot {^{12}C_K} \cdot {^{11}C_{K - 1}}} $ is equal to $\frac{{12 \times 21 \times 19 \times 17 \times \dots \times 3}}{{11!}} \times {2^{12}} \times p$,then $p$ is:

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In the expansion of $(1+x)^n$,the sum $\frac{C_1}{C_0} + 2 \frac{C_2}{C_1} + 3 \frac{C_3}{C_2} + \ldots + n \frac{C_n}{C_{n-1}}$ is equal to:

MediumKCET 2024

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

DifficultJEE MAIN 2022

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If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2 + ...... + ^{2017}C_{1008} = \lambda^2$ where $\lambda > 0$,then the remainder when $\lambda$ is divided by $33$ is:

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The sum of the series $aC_0 + (a + b)C_1 + (a + 2b)C_2 + \dots + (a + nb)C_n$ is,where $C_r$ denotes the combinatorial coefficient in the expansion of $(1 + x)^n, n \in N$.

Similar Questions

$3 \cdot C_0 + 7 \cdot C_1 + 11 \cdot C_2 + \ldots + (3 + 4n) C_n =$

If $\sum\limits_{K = 1}^{12} {12K \cdot {^{12}C_K} \cdot {^{11}C_{K - 1}}} $ is equal to $\frac{{12 \times 21 \times 19 \times 17 \times \dots \times 3}}{{11!}} \times {2^{12}} \times p$,then $p$ is:

In the expansion of $(1+x)^n$,the sum $\frac{C_1}{C_0} + 2 \frac{C_2}{C_1} + 3 \frac{C_3}{C_2} + \ldots + n \frac{C_n}{C_{n-1}}$ is equal to:

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

If $^{2017}C_0 + ^{2017}C_1 + ^{2017}C_2 + ...... + ^{2017}C_{1008} = \lambda^2$ where $\lambda > 0$,then the remainder when $\lambda$ is divided by $33$ is:

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