If the value of C_{0}+2 cdot C_{1}+3 cdot C_{ | vedclass

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(A) We are given the sum $S = \sum_{k=0}^{n} (k+1) C_{k} = 576$.We know that $\sum_{k=0}^{n} C_{k} x^{k} = (1+x)^{n}$.Multiplying by $x$,we get $\sum_

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$7$

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(A) We are given the sum $S = \sum_{k=0}^{n} (k+1) C_{k} = 576$.We know that $\sum_{k=0}^{n} C_{k} x^{k} = (1+x)^{n}$.Multiplying by $x$,we get $\sum_{k=0}^{n} C_{k} x^{k+1} = x(1+x)^{n}$.Differentiating with respect to $x$: $\sum_{k=0}^{n} (k+1) C_{k} x^{k} = (1+x)^{n} + nx(1+x)^{n-1}$.Setting $x=1$,we get $\sum_{k=0}^{n} (k+1) C_{k} = (1+1)^{n} + n(1)(1+1)^{n-1} = 2^{n} + n \cdot 2^{n-1}$.This simplifies to $2^{n-1}(2+n) = 576$.We can write $576 = 64 	imes 9 = 2^{6} 	imes 9 = 2^{7-1}(7+2)$.Comparing $2^{n-1}(n+2) = 2^{7-1}(7+2)$,we get $n=7$.

If the value of $C_{0}+2 \cdot C_{1}+3 \cdot C_{2}+\ldots+(n+1) \cdot C_{n}=576$,then $n$ is equal to

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