(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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Binomial Theorem

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Properties of binomial coefficients

Let $m, n \in \mathbb{N}$ and $\operatorname{gcd}(2, n)=1$. If $30\binom{30}{0} + 29\binom{30}{1} + \ldots + 2\binom{30}{28} + 1\binom{30}{29} = n \cdot 2^m$,then $n + m$ is equal to (Here $\binom{n}{k} = {^nC_k}$)

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If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .......... + C_nx^n,$ then $C_0^2 + C_1^2 + C_2^2 + C_3^2 + ...... + C_n^2$ =

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The sum of the last eight coefficients in the expansion of $(1 + x)^{15}$ is

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If $\frac{1}{n+1} {}^{n}C_{n} + \frac{1}{n} {}^{n}C_{n-1} + \dots + \frac{1}{2} {}^{n}C_{1} + {}^{n}C_{0} = \frac{1023}{10}$,then $n$ is equal to

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If ${ }^{n} C_0+\frac{1}{2}{ }^{n} C_1+\frac{1}{3}{ }^{n} C_2+\ldots+\frac{1}{n+1}{ }^{n} C_{n}=\frac{1023}{10}$,then $n=$

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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

Let $m, n \in \mathbb{N}$ and $\operatorname{gcd}(2, n)=1$. If $30\binom{30}{0} + 29\binom{30}{1} + \ldots + 2\binom{30}{28} + 1\binom{30}{29} = n \cdot 2^m$,then $n + m$ is equal to (Here $\binom{n}{k} = {^nC_k}$)

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .......... + C_nx^n,$ then $C_0^2 + C_1^2 + C_2^2 + C_3^2 + ...... + C_n^2$ =

The sum of the last eight coefficients in the expansion of $(1 + x)^{15}$ is

If $\frac{1}{n+1} {}^{n}C_{n} + \frac{1}{n} {}^{n}C_{n-1} + \dots + \frac{1}{2} {}^{n}C_{1} + {}^{n}C_{0} = \frac{1023}{10}$,then $n$ is equal to

If ${ }^{n} C_0+\frac{1}{2}{ }^{n} C_1+\frac{1}{3}{ }^{n} C_2+\ldots+\frac{1}{n+1}{ }^{n} C_{n}=\frac{1023}{10}$,then $n=$

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