C02+C16+C212+.......+Cn(n+1)(n+2)=

C0x+C12x2+C23x3+.......+Cnn+1xn+1=(1+x)n+1−1(n+1)Integrating on b.s, C0x22+C12x33+.......+Cnn+1xn+2n+2=(1+x)n+2(n+1)(n+2)−xn+1+KPut x = 0 , ⇒K=−1(n+1)(n+2)∴C0x22+C16x3+....+Cn(n+1)(n+2)xn+2=(1+x)n+2−(n+2)x−1(n+1)(n+2)Put x = 1,C02+C16+.......+Cn(n+1)(n+2)=2n+2−n−3(n+1)(n+2)

C02+C16+C212+.......+Cn(n+1)(n+2)=

C0x+C12x2+C23x3+.......+Cnn+1xn+1=(1+x)n+1−1(n+1)Integrating on b.s, C0x22+C12x33+.......+Cnn+1xn+2n+2=(1+x)n+2(n+1)(n+2)−xn+1+KPut x = 0 , ⇒K=−1(n+1)(n+2)∴C0x22+C16x3+....+Cn(n+1)(n+2)xn+2=(1+x)n+2−(n+2)x−1(n+1)(n+2)Put x = 1,C02+C16+.......+Cn(n+1)(n+2)=2n+2−n−3(n+1)(n+2)

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$\frac{C_{0}}{2} + \frac{C_{1}}{6} + \frac{C_{2}}{12} + ....... + \frac{C_{n}}{(n + 1) (n + 2)} =$

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

$\frac{2^{n + 2} + n + 1}{(n + 1) (n + 2)}$

$\frac{2^{n + 2} - n - 3}{(n + 1) (n + 2)}$

$\frac{2^{n + 1} + n + 3}{(n + 1) (n + 2)}$

answer is C.

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Detailed Solution

$C_{0} x + \frac{C_{1}}{2} x^{2} + \frac{C_{2}}{3} x^{3} + ....... + \frac{C_{n}}{n + 1} x^{n + 1} = \frac{{(1 + x)}^{n + 1} - 1}{(n + 1)}$

Integrating on b.s, $C_{0} \frac{x^{2}}{2} + \frac{C_{1}}{2} \frac{x^{3}}{3} + ....... + \frac{C_{n}}{n + 1} \frac{x^{n + 2}}{n + 2} = \frac{{(1 + x)}^{n + 2}}{(n + 1) (n + 2)} - \frac{x}{n + 1} + K$

Put x = 0 , $\Rightarrow K = \frac{- 1}{(n + 1) (n + 2)}$

$∴ C_{0} \frac{x^{2}}{2} + \frac{C_{1}}{6} x^{3} + .... + \frac{C_{n}}{(n + 1) (n + 2)} x^{n + 2} = \frac{{(1 + x)}^{n + 2} - (n + 2) x - 1}{(n + 1) (n + 2)}$