Find the sum of the series - r-0 n -1 r - n C r 1 2 r - 3 r 2 2r - 7 r 2 3r - 15 r 2 4r -upto m terms

∑ _ r=0 ^ n ( 1 ) #160;^ r ^ n C _ r { 1 2 ^ r + 3 ^ r 2 ^ 2r + 7 ^ r 2 ^ 3r + 15 ^ r 2 ^ 4r +.... upto m terms #160;} #160; =∑ _ ...

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Byju's Answer

Standard XII

Mathematics

Sum of Coefficients of All Terms

Find the sum ...

Question

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

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Solution

$\sum_{r = 0}^{n} {(- 1)}^{r} {^{n} C}_{r} {\frac{1}{2^{r}} + \frac{3^{r}}{2^{2 r}} + \frac{7^{r}}{2^{3 r}} + \frac{15^{r}}{2^{4 r}} + . . . . upto m terms}$
$= n \sum r = 0 {(- 1)}^{r} {^{n} C}_{r} {\frac{1}{2^{r}} + \frac{3^{r}}{4^{r}} + \frac{7^{r}}{8^{r}} + \frac{15^{r}}{16^{r}} + . . . . . upto m terms}$
$= \sum_{r = 0}^{n} {(- 1)}^{r} {^{n} C}_{r} {{(\frac{1}{2})}^{r} + {(\frac{3}{4})}^{r} + {(\frac{7}{8})}^{r} + {(\frac{15}{16})}^{r} + . . . . upto m terms}$
$= n \sum r = 0 {(- 1)}^{r} {^{n} C}_{r} {(\frac{1}{2})}^{r} + n \sum r = 0 {(- 1)}^{r} {^{n} C}_{r} {(\frac{3}{4})}^{r} + n \sum r = 0 {(- 1)}^{r} {^{n} C}_{r} {(\frac{7}{8})}^{r} + n \sum r = 0 {(- 1)}^{r} {^{n} C}_{r} {(\frac{15}{16})}^{r}$
$+ . . . . . upto m terms$
$= {(1 - \frac{1}{2})}^{n} + {(1 - \frac{3}{4})}^{n} + {(1 - \frac{7}{8})}^{n} + {(1 - \frac{15}{16})}^{n} + . . . . . . upto m terms$
$= \frac{1}{2^{n}} + \frac{1}{4^{n}} + \frac{1}{8^{n}} + . . . . . upto m terms$
$= \frac{\frac{1}{2^{n}} (1 - {(\frac{1}{2^{n}})}^{m})}{1 - \frac{1}{2^{n}}}$
$= \frac{2^{m n} - 1}{2^{m n} (2^{n} - 1)}$

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Similar questions

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}} + . . . 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]

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

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}} + \dots \infty]

is equal to

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}} + \dots \infty]

is equal to

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Find the sum of the series∑nr=0(−1)r.nCr{12r+3r22r+7r23r+15r24r+....uptomterms}

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