The sum of the series -r - 0n -1r nCr 12r - 3r22r - 7r23r - 15r24r - - m terms is

The correct option is A 2^mn 12^mn(2^n 1) ∑_r = 0^n ( 1)^r ^nC_r (12^r + 3^r2^2r + 7^r2^3r + ....upto m terms ) = ∑_r = 0^n ( 1)^r ^nC_ ...

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Byju's Answer

Standard XII

Mathematics

Multinomial Expansion

The sum of th...

Question

The sum of the series $n \sum r = 0 (- 1)^{r}^{n} C_{r} (\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . m t e r m s)$ is

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

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\frac{2^{m n} - 1}{2^{n} - 1}

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\frac{2^{m n} + 1}{2^{n} + 1}

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None of these

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Solution

The correct option is A $\frac{2^{m n - 1}}{2^{m n} (2^{n} - 1)}$
$n \sum r = 0 (- 1)^{r}^{n} C_{r}$
$(\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + . . . . u p t o m t e r m s)$
$= n \sum r = 0 (- 1)^{r}^{n} C_{r} \frac{1}{2^{r}} + n \sum r = 0 (- 1)^{r}^{n} C_{r} . \frac{3^{r}}{2^{2 r}} + n \sum r = 0 (- 1)^{r}^{n} C_{r} . \frac{7^{r}}{2^{3 r}} + . . .$
$= {(1 - \frac{1}{2})}^{n} + {(1 - \frac{3}{4})}^{n} + {(1 - \frac{7}{8})}^{n} + . . . .$ upto $m$ terms
$= \frac{1}{2^{n}} + \frac{1}{4^{n}} + \frac{1}{8^{n}} + . . . . .$ upto m terms
$= \frac{\frac{1}{2^{n}} (1 - \frac{1}{2^{m n}})}{(1 - \frac{1}{2^{n}})} = \frac{2^{m n} - 1}{2^{m n} (2^{n} - 1)}$

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Similar questions

Find the sum of the series
$\sum_{n}^{r = 0} (- 1)^{r}^{n} C_{r} [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} \dots u p t o m t e r m s]$

Q. The sum of the series

n \sum r = 0 {(- 1)}^{r}^{n} C_{r} [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + \dots u p t o p t e r m s]

Q. Find the sum of the series

\sum_{r = 0}^{n} {(- 1)}^{r} . {^{n} C}_{r} {\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . . u p t o m t e r m s}

Find the sum of the series
$\sum_{n}^{r = 0} (- 1)^{r}^{n} C_{r} [\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} \dots u p t o m t e r m s]$

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The sum of the series n∑r=0(−1)r nCr(12r+3r22r+7r23r+15r24r+...m terms) is

The correct option is A 2mn−12mn(2n−1)n∑r=0(−1)r nCr(12r+3r22r+7r23r+....upto m terms)=n∑r=0(−1)r nCr12r+n∑r=0(−1)r nCr.3r22r+n∑r=0(−1)r nCr.7r23r+...=(1−12)n+(1−34)n+(1−78)n+.... upto m terms=12n+14n+18n+..... upto m terms=12n(1−12mn)(1−12n)=2mn−12mn(2n−1)

The sum of the series $n \sum r = 0 (- 1)^{r}^{n} C_{r} (\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . m t e r m s)$ is