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Sum of Coefficients of All Terms

If C0,C1,C2...

Question

If $C_{0}, C_{1}, C_{2}, . . . . ., C_{n}$ are Binomial coefficients then find the value of $C_{0} + 2. C_{1} + 3. C_{2} + . . . . . + (n + 1) . C_{n} .$

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Solution

$T_{r} = (r + 1)^{n} C_{r}$
$\sum T_{r} = \sum_{r = 0}^{n} (r + 1)^{n} C_{r} = C_{0} + 2 C_{1} + 3 C_{2} . . . . (n + 1) C_{n}$
$= \sum_{r = 0}^{n} r .^{n} C_{n} + \sum_{r = 0}^{n}$ $^{n} C_{r}$
$= \sum_{r = 0}^{n} r \times {\frac{n}{r}}^{n - 1} C_{r - 1} + \sum_{r = 0}^{n}$ $^{n} C_{r}$ $[^{n} C_{r} = {\frac{n}{r}}^{n - 1} C_{r - 1}]$
$= n \times 2^{n - 1} + 2^{n}$ $[\sum^{n - 1} C_{r - 1} = 2^{n - 1}$ and $\sum^{n} C_{r} = 2^{n}]$
$= [\frac{n + 2}{2}] 2^{n} - (n + 2) 2^{n - 1}$

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C_{0}, C_{1}, C_{2}, . . . . . . . . . . C_{n}

are the binomial coefficients in the expansion of

(1 + x)^{n}

. n being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2}, \dots, C_{n}

are Binomial Coefficient in the expansion of

{(1 + x)}^{n}

C_{0} + \frac{C_{1}}{2} + \frac{C_{2}}{3} + \dots + \frac{C_{n}}{n + 1}

C_{0}, C_{1}, C_{2}, . . . . . . . . . . . C_{n}

are the Binomial coefficients in the expansion

(1 + x)^{n} .

‘n’ being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . . . (C_{0} + C_{1} + C_{2} + . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2} . . . . . . . . . . ., C_{n}

are the binomial coefficient in the expansion of

(1 + x)^{n}, n

being even, then

C_{0} + (C_{0} + C_{1}) + (C_{0} + C_{1} + C_{2}) + . . . . . . . + (C_{0} + C_{1} + C_{2} + . . . . . . . . + C_{n - 1})

C_{0}, C_{1}, C_{2} . . . ., C_{n}

denote the binomial coefficients in the expansion of

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\frac{C 1}{C 0} + 2 \frac{C 2}{C 1} + + 3 \frac{C 3}{C 2} + . . . . . + n \frac{C n}{C n - 1}

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