The value of the expression C1-C0-2C2-C1-3C3-C2- -nCn-Cn-1 is

The correct option is A n(n+1)2 Given: C_1C_0 + 2C_2C_1+3C_3C_2 +⋯+nC_nC_n 1 r^th term of the above expression is T_r = rC_rC_r 1 = rn r+1r ...

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Standard XII

Mathematics

General Term of Binomial Expansion

The value of ...

Question

The value of the expression $\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots \dots \dots + n \frac{C_{n}}{C_{n - 1}}$ is

\frac{(n + 1) (n + 2)}{2}

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\frac{n (n + 1)}{2}

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\frac{n (n - 1)}{2}

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\frac{n (n + 2)}{2}

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Solution

The correct option is A $\frac{n (n + 1)}{2}$
Given:
$\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots + n \frac{C_{n}}{C_{n - 1}}$

$r^{t h}$ term of the above expression is
$T_{r} = r \frac{C_{r}}{C_{r - 1}} = r \frac{n - r + 1}{r} = n - r + 1$

Now,
$n \sum r = 1 T_{r} = n \sum r = 1 (n - r + 1)$

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

$= \frac{n (n + 1)}{2}$

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Similar questions

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

\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}

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

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

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}

Q. Prove that the sum of the series

\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}

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The value of the expression C1C0+2C2C1+3C3C2+………+nCnCn−1 is

The correct option is A n(n+1)2Given:C1C0+2C2C1+3C3C2+⋯+nCnCn−1rth term of the above expression isTr=rCrCr−1=rn−r+1r=n−r+1Now,n∑r=1Tr=n∑r=1(n−r+1) =n2−n(n+1)2+n=n(n+1)2

The value of the expression $\frac{C_{1}}{C_{0}} + 2 \frac{C_{2}}{C_{1}} + 3 \frac{C_{3}}{C_{2}} + \dots \dots \dots + n \frac{C_{n}}{C_{n - 1}}$ is