(D) The general term of the series is given by $\frac{r \cdot {}^nC_r}{{}^nC_{r-1}}$.Using the formula ${}^nC_r = \frac{n!}{r!(n-r)!}$,we have:$\frac{

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(D) The general term of the series is given by $\frac{r \cdot {}^nC_r}{{}^nC_{r-1}}$.Using the formula ${}^nC_r = \frac{n!}{r!(n-r)!}$,we have:$\frac{r \cdot {}^nC_r}{{}^nC_{r-1}} = \frac{r \cdot \frac{n!}{r!(n-r)!}}{\frac{n!}{(r-1)!(n-r+1)!}} = \frac{r \cdot (r-1)!(n-r+1)!}{r!(n-r)!} = \frac{n-r+1}{1} = n-r+1$.For $r=1$,the term is $n-1+1 = n$.For $r=2$,the term is $n-2+1 = n-1$.Continuing this,for $r=n$,the term is $n-n+1 = 1$.Thus,the sum is $n + (n-1) + (n-2) + \ldots + 1 = \sum_{k=1}^{n} k$.

Class 11 Mathematics Binomial Theorem General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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If $C_j$ stands for ${}^nC_j$,then $\frac{C_1}{C_0} + \frac{2 \times C_2}{C_1} + \frac{3 \times C_3}{C_2} + \ldots + \frac{n \times C_n}{C_{n-1}} = $

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$\sum_{k=1}^{n} k^2$
B
$\sum_{k=1}^{n} \frac{k}{2}$
C
$\sum_{k=1}^{n} 2k$
D
$\sum_{k=1}^{n} k$

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Binomial Theorem

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General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

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If $C_j$ stands for ${}^nC_j$,then $\frac{C_1}{C_0} + \frac{2 \times C_2}{C_1} + \frac{3 \times C_3}{C_2} + \ldots + \frac{n \times C_n}{C_{n-1}} = $

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If the coefficients of the three consecutive terms in the expansion of $(1+x)^n$ are in the ratio $1:5:20$,then the coefficient of the fourth term is $............$.

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