The sum of the series 30-6-562-52-302-62-5263-53-303-63-5364-54- is

T_n=30^n(6^n 5^n)(6^n+1 5^n+1) =6^n · 5^n(6 5)(6^n 5^n)(6^n+1 5^n+1) =6^n+15^n 5^n+16^n(6^n 5^n)(6^n+1 5^n+1) =6^n+15^n 6^2n+1+6^2n+1 5^n+16^n(6^n 5^n)(6^n+1 ...

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Standard XII

Mathematics

Sum of n Terms

The sum of th...

Question

The sum of the series
$\frac{30}{(6 - 5) (6^{2} - 5^{2})} + \frac{30^{2}}{(6^{2} - 5^{2}) (6^{3} - 5^{3})} + \frac{30^{3}}{(6^{3} - 5^{3}) (6^{4} - 5^{4})} + \dots \infty$ is

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Solution

$T_{n} = \frac{30^{n}}{(6^{n} - 5^{n}) (6^{n + 1} - 5^{n + 1})} = \frac{6^{n} \cdot 5^{n} (6 - 5)}{(6^{n} - 5^{n}) (6^{n + 1} - 5^{n + 1})} = \frac{6^{n + 1} 5^{n} - 5^{n + 1} 6^{n}}{(6^{n} - 5^{n}) (6^{n + 1} - 5^{n + 1})} = \frac{6^{n + 1} 5^{n} - 6^{2 n + 1} + 6^{2 n + 1} - 5^{n + 1} 6^{n}}{(6^{n} - 5^{n}) (6^{n + 1} - 5^{n + 1})} = \frac{6^{n}}{6^{n} - 5^{n}} - \frac{6^{n + 1}}{6^{n + 1} - 5^{n + 1}}$

$S_{n} = T_{1} + T_{2} + \dots + T_{n} = (\frac{6}{6 - 5} - \frac{6^{2}}{6^{2} - 5^{2}}) + (\frac{6^{2}}{6^{2} - 5^{2}} - \frac{6^{3}}{6^{3} - 5^{3}}) + \dots + (\frac{6^{n}}{6^{n} - 5^{n}} - \frac{6^{n + 1}}{6^{n + 1} - 5^{n + 1}}) = \frac{6}{6 - 5} - \frac{6^{n + 1}}{6^{n + 1} - 5^{n + 1}} ∴ S_{\infty} = 6 - lim n \to \infty \frac{6^{n + 1}}{6^{n + 1} - 5^{n + 1}} = 6 - lim n \to \infty \frac{1}{1 - {(\frac{5}{6})}^{n + 1}} = 6 - \frac{1}{1 - 0} = 6 - 1 = 5$

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Q. The sum of the infinite series

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Q. Find the sum of the infinite series

1^{2} - \frac{2^{2}}{5} + \frac{3^{2}}{5^{2}} - \frac{4^{2}}{5^{3}} + \frac{5^{2}}{5^{4}} - \frac{6^{2}}{5^{5}} + . . . .

Q. Prepare a discrete series from the following data:

62	50	57	58	51	53	62	64	60	61
60	51	64	55	55	52	60	65	58	60
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Q. The sum of the series

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(\frac{61}{30})^{3} - (\frac{5}{6})^{3} - (\frac{6}{5})^{3}

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62	50	57	58	51	53	62	64	60	61
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59	52	63	56	56	58	64	63	62	60

62	50	57	58	51	53	62	64	60	61
60	51	64	55	55	52	60	65	58	60
59	52	63	56	56	58	64	63	62	60

The sum of the series 30(6−5)(62−52)+302(62−52)(63−53)+303(63−53)(64−54)+⋯∞ is

The sum of the series
$\frac{30}{(6 - 5) (6^{2} - 5^{2})} + \frac{30^{2}}{(6^{2} - 5^{2}) (6^{3} - 5^{3})} + \frac{30^{3}}{(6^{3} - 5^{3}) (6^{4} - 5^{4})} + \dots \infty$ is

62	50	57	58	51	53	62	64	60	61
60	51	64	55	55	52	60	65	58	60
59	52	63	56	56	58	64	63	62	60