The sum of the series ∑r=0n (−1)r  nCr12r+3r22r+7r23r+15r24r+…+m terms  is

# The sum of the series

1. A

$\frac{{2}^{\mathrm{mn}}-1}{{2}^{\mathrm{mn}}\left({2}^{\mathrm{n}}-1\right)}$

2. B

$\frac{{2}^{\mathrm{mn}}-1}{{2}^{\mathrm{n}}-1}$

3. C

$\frac{{2}^{\mathrm{mn}}+1}{{2}^{\mathrm{n}}+1}$

4. D

None of these

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### Solution:

$+\sum _{\mathrm{r}=0}^{\mathrm{n}} \left(-1{\right)}^{\mathrm{rn}}{\mathrm{C}}_{\mathrm{r}}\frac{{7}^{\mathrm{r}}}{{2}^{3\mathrm{r}}}+\dots$

$={\left(1-\frac{1}{2}\right)}^{\mathrm{n}}+{\left(1-\frac{3}{4}\right)}^{\mathrm{n}}+{\left(1-\frac{7}{8}\right)}^{\mathrm{n}}+\dots$up to M terms

$=\frac{\frac{1}{{2}^{\mathrm{n}}}\left\{1-{\left(\frac{1}{{2}^{\mathrm{n}}}\right)}^{\mathrm{m}}\right\}}{\left(1-\frac{1}{{2}^{\mathrm{n}}}\right)}=\frac{{2}^{\mathrm{mn}}-1}{{2}^{\mathrm{mn}}\left({2}^{\mathrm{n}}-1\right)}$

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