The sum of the series -r-1n-1r-1- nCra-r is equal to

The correct option is A a Simplifying, we get ∑ ( 1)^r 1(a) ^nC_r+∑ ( 1)^rr ^nC_r Expanding, we get a[ ^nC_1 ^nC_2+...( 1)^n ^nC_n]+[ ^nC_1+2 ^n ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

The sum of th...

Question

The sum of the series $\sum_{r = 1}^{n} (- 1)^{r - 1} \cdot^{n} C_{r} (a - r)$ is equal to

n {.2}^{n - 1} + a

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Solution

The correct option is A $a$
Simplifying, we get
$\sum (- 1)^{r - 1} (a)^{n} C_{r} + \sum (- 1)^{r} r^{n} C_{r}$
Expanding, we get
$a [^{n} C_{1} -^{n} C_{2} + . . . (- 1)^{n}^{n} C_{n}] + [-^{n} C_{1} + 2^{n} C_{2} - 3^{n} C_{3} + . . . (- 1)^{n} C_{n}]$
$= a [^{n} C_{1} -^{n} C_{2} + . . . (- 1)^{n}^{n} C_{n}] - [^{n} C_{1} - 2^{n} C_{2} + 3^{n} C_{3} + . . . (- 1)^{n} C_{n}]$
$= a [- (1 + x)^{n} |_{x = - 1} + 1] - [\frac{d (1 + x)^{n}}{d x} |_{x = - 1}]$
$= a [0 + 1] - [0]$
$= a$

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