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Relation between Coefficient and Indices of x and y

If C0, C1,...

Question

If $C_{0}, C_{1}, C_{2}, . . . . . . C_{15}$ are the binomial coefficients in the expansion of $(1 + x)^{15}$ , prove that
$\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + . . . . . . . . . . . + 15 \frac{C_{15}}{C_{14}} = 120$

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Solution

Putting n = 15 in part (a),we get

$L . H . S . = \frac{15.16}{2} = 120 = R . H . S$

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Similar questions

C_{0}, C_{1}, C_{2}, \dots, C_{15}

are binomial coefficients in

{(1 + x)}^{15}

\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots + 15 \frac{C_{15}}{C_{14}}

c_{0}, c_{1}, c_{2}, \dots c_{15}

are the Binomial co-efficients in the expansion of

(1 + x)^{15}

, then the value of

\frac{c_{1}}{c_{0}} + 2 \frac{c_{2}}{c_{1}} + 3 \frac{c_{3}}{c_{2}} + \dots + 15 \frac{c_{15}}{c_{14}}

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

are the binomial coefficients in the expansion of

(1 + x)^{n}

then prove that:

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

\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + . . . . + 15 \frac{C_{15}}{C_{14}}

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

denote the binomial coefficients in the expansion of

{(1 + x)}^{n}

\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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