Prove that 22C01-2-23C12-3-24C23-4-2n-2Cnn-1n-2-3n-2-2n-5n-1n-2

Let S= 2 ^ 2 C _ 0 1.2 + 2 ^ 3 C _ 1 2.3 +.....+ 2 ^ n+2 C _ n ( n+1 ) ( n+2 ) #160; S=∫[ ∫( 1+x ) #160; ^ n dx #160; ] dx at x=2 ...

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nCr Definitions and Properties

Prove that ...

Question

Prove that $2^{2} \frac{C_{0}}{1.2} + 2^{3} \frac{C_{1}}{2.3} + 2^{4} \frac{C_{2}}{3.4} + . . . . . . . . . + 2^{n + 2} \frac{C_{n}}{(n + 1) (n + 2)} = \frac{3^{n + 2} - 2 n - 5}{(n + 1) (n + 2)}$

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(1 + x)^{n} = n \sum r = 0 C_{r} x^{r}

\frac{2^{2} C_{0}}{1.2} + \frac{2^{3} C_{1}}{2.3} + \frac{2^{4} C_{2}}{3.4} + \dots + \frac{2^{n + 2} C_{n}}{(n + 1) (n + 2)}

(1 + x)^{n} = \sum_{r = 0}^{n} C_{r} x^{r}

\frac{2^{2} C_{0}}{1.2} + \frac{2^{3} C_{1}}{2.3} + . . . . . . + \frac{2^{n + 2} C_{n}}{(n + 1) (n + 2)}

(1 + x)^{n} = n \sum r = 0^{n} C_{r} x^{n},

\frac{C_{0}}{1 \cdot 2} 2^{2} + \frac{C_{1}}{2 \cdot 3} 2^{3} + \frac{C_{2}}{3 \cdot 4} 2^{4} + \dots + \frac{C_{n}}{(n + 1) (n + 2)} 2^{n + 2}

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n - \frac{n (n - 1) (n - 2)}{⌊ 3} \cdot 3 + \frac{n (n - 1) (n - 2) (n - 3) (n - 4)}{⌊ 5} \cdot 3^{2} - . . . . .,

n - \frac{n (n - 1) (n - 2)}{⌊ 3} \cdot \frac{1}{3} + \frac{n (n - 1) (n - 2) (n - 3) (n - 4)}{⌊ 5} \cdot \frac{1}{3^{2}} - . . . . .,

\frac{C_{0}}{1.2} + \frac{C_{1}}{2.3} + \frac{C_{2}}{3.4} + . . . . . . . . . . . + \frac{C_{n}}{(n + 1) (n + 2)} =

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