Find C1C0-2C2C1-3C3C2-nCnCn-1-

S=^nC_1^nC_0+2^nC_2^nC_1+3^nC_3^nC_2+......+n^nC_n^nC_n 1 S=∑_r=1^n(r^nC_r^nC_r 1)........(1) Now, ^nC_r^nC_r 1=n!r!(n r)!n!(r 1)!(n r+1)! =( ...

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Byju's Answer

Standard XII

Mathematics

Binomial Theorem for Any Index

Find C1C0+2...

Question

Find $\frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . + \frac{n C_{n}}{C_{n - 1}} =$

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Solution

$S = \frac{^{n} C_{1}}{^{n} C_{0}} + 2 \frac{^{n} C_{2}}{^{n} C_{1}} + 3 \frac{^{n} C_{3}}{^{n} C_{2}} + . . . . . . + n \frac{^{n} C_{n}}{^{n} C_{n - 1}}$
$S = \sum_{r = 1}^{n} (r \frac{^{n} C_{r}}{^{n} C_{r - 1}}) . . . . . . . . (1)$
Now,
$\frac{^{n} C_{r}}{^{n} C_{r - 1}} = \frac{\frac{n!}{r! (n - r)!}}{\frac{n!}{(r - 1)! (n - r + 1)!}}$
$= \frac{(r - 1)! (n - r + 1)!}{r! (n - r)!}$
$= \frac{(r - 1)! (n - r + 1) (n - r)!}{r (r - 1)! (n - r)!}$
$= \frac{n - r + 1}{r}$
Hence (1) become
$S = \sum_{r = 1}^{n} (r \frac{n - r + 1}{r})$
$= \sum_{r = 1}^{n} (n - r + 1)$
$= \sum_{r = 1}^{n} n + \sum_{r = 1}^{n} 1 - \sum_{r = 1}^{n} r$
$= n (1 + 1 + 1..... n) + (1 + 1 + 1..... n) - (1 + 2 + 3 + . . . . + n)$
$= n (n) + n - \frac{n (n + 1)}{2}$
$= (n^{2} + n) - \frac{n^{2} + n}{2}$
$= \frac{1}{2} (n^{2} + n)$
$∴ \frac{^{n} C_{1}}{^{n} C_{0}} + 2 \frac{^{n} C_{2}}{^{n} C_{1}} + 3 \frac{^{n} C_{3}}{^{n} C_{2}} + . . . . . . + n \frac{^{n} C_{n}}{^{n} C_{n - 1}} = \frac{1}{2} (n^{2} + n)$

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Similar questions

\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}

Q. Solve

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

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

i f (1 + x)^{n} = C_{0} + c_{1} x + . . . + C_{n} x^{n} +, t h e n \frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . . + \frac{n C_{n}}{C_{n - 1}} i s

Q. The value of the expression

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

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Find C1C0+2C2C1+3C3C2+....+nCnCn−1=

Find $\frac{C_{1}}{C_{0}} + \frac{2 C_{2}}{C_{1}} + \frac{3 C_{3}}{C_{2}} + . . . . + \frac{n C_{n}}{C_{n - 1}} =$