Find $C_{0} +^{2} C_{1} +^{3} C_{2} +^{4} C_{3} + . . . . . . . . . . . +^{(n + 1)} C_{n}$

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Solution

$C_{0} +^{2} C_{1} +^{3} C_{2} +^{4} C_{3} + . . . . . . . . . . . +^{(n + 1)} C_{n}$
So, we know that
${(1 + x)}^{2} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + C_{4} x^{4} + . . . . . . . + C_{n} x^{n} - - - - (1)$
$n . {(1 + n)}^{n - 1} = 0 + C_{1} + 2 C_{2} x + 3 C_{3} x^{2} + . . . . . . . . . + n C_{n} x^{(n - 1)}$
diff. with respect to x
$n . {(1 + n)}^{n - 1} = 0 + C_{1} + 2 C_{2} x + 3 C_{3} x^{2} + . . . . . . . . . + n C_{n} x^{(n - 1)}$
put $x = 1$ in eq(1) and eq(2)
$2^{n} + 2 (2^{n - 1}) = C_{0} + 2 C_{1} + 3 C_{2} + 4 C_{3} + . . . . . . . + (n + 1) C_{n}$
$2^{n} + n {.2}^{n - 1}$ Required value

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