Find the sum 1 2 3 ⋅ 1 3 + 1 2 + 2 2 5 ( 1 3 + 2 3 ) + 1 2 + 2 2 + 3 2 7 ( 1 3 + 2 3 + 3 3 ) + ⋯ up to the 10th term.

The n th term is ( 2 n + 1 ) n ( n + 1 ) ( 2 n + 1 ) /6 n 2 ( n + 1 ) 2 /4 = 3 n ( n + 1 ) /2 . Thus S 10 = 2 3 ( 385 + 55 ) = 660 .

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Mathematics - Sequence & Series Question with Solution | TestHub

MathematicsSequence & SeriesSpecial sequences/seriesMedium2 minQB

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Find the sum $\frac{3 \cdot 1 ^{3}}{1 ^{2}} + \frac{5 ( 1 ^{3} + 2 ^{3} )}{1 ^{2} + 2 ^{2}} + \frac{7 ( 1 ^{3} + 2 ^{3} + 3 ^{3} )}{1 ^{2} + 2 ^{2} + 3 ^{2}} + \dots$ up to the 10th term.

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Solution:

The $n$ th term is $(2 n + 1) \frac{n ^{2} ( n + 1 ) ^{2} /4}{n ( n + 1 ) ( 2 n + 1 ) /6} = 3 n (n + 1) /2$ . Thus $S_{10} = \frac{3}{2} (385 + 55) = 660$ .

Stream:JEESubject:MathematicsTopic:Sequence & SeriesSubtopic:Special sequences/series

⏱ 2mℹ️ Source: QB

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