If (1 + x)^n = C_0 + C_1x + C_2x^2 + .... + C | vedclass

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Question

(A) Given the expansion $(1 + x)^n = \sum_{r=0}^{n} C_r x^r$.We need to find the sum $S = \sum_{r=0}^{n} (r + 1) C_r$.This can be written as $S = \sum

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$(n + 2)2^{n - 1}$

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(A) Given the expansion $(1 + x)^n = \sum_{r=0}^{n} C_r x^r$.We need to find the sum $S = \sum_{r=0}^{n} (r + 1) C_r$.This can be written as $S = \sum_{r=0}^{n} r C_r + \sum_{r=0}^{n} C_r$.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,$S = n 2^{n-1} + 2^n$.$S = n 2^{n-1} + 2 \cdot 2^{n-1} = (n + 2) 2^{n-1}$.Alternatively,for $n=1$,the expression is $C_0 + 2C_1 = 1 + 2(1) = 3$. Checking the options for $n=1$: $(1+2)2^{1-1} = 3(1) = 3$. Thus,option $(A)$ is correct.

If $(1 + x)^n = C_0 + C_1x + C_2x^2 + .... + C_nx^n$,then the value of $C_0 + 2C_1 + 3C_2 + .... + (n + 1)C_n$ will be

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