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Relation between Coefficient and Indices of x and y

If C0, C1, ...

Question

If $C_{0}, C_{1}, C_{2}, \dots C_{n}$ are coefficients of expansion $(1 + x)^{n}$ then prove that:
$C_{0} + 3. C_{1} + 5. C_{2} + \dots + (2 n + 1) . C_{n} = (n + 1) 2^{n}$ .

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Solution

L.H.S.
$=^{n} C_{0} + 3. C_{1} + 5. C_{2} + . . . + (2 n + 1)^{n} C_{n}$

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

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

$= 2.^{n} C_{1} + 4^{n} C_{2} + 6^{n} C_{3} + . . . + 2 n^{n} C_{n} + (^{n} C_{0} +^{n} C_{1} +^{n} C_{2} +^{n} C_{3} + . . .^{n} C_{n})$

$= 2 [^{n} C_{1} + 2^{n} C_{2} + 3.^{n} C_{3} + . . . +^{n} C_{n}] + (1 + 1) n$

$= 2 [n + \frac{2 n . (n - 1)}{2!} + \frac{3 n . (n - 1) (n - 2)}{3!} + . . . + n t e r m s] + 2^{n}$

$= 2 n [1 + (n - 1) + \frac{(n - 1) (n - 2)}{2!} + . . . + n t e r m s] + 2^{n}$

$= 2 n . (1 + 1)^{n - 1} + 2^{n} = 2 n {.2}^{n - 1} + 2^{n} + n {.2}^{n} + 2^{n}$

$= (n + 1) 2^{n} = R . H . S .$

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