The sum to n terms of the series 312-512-22-712-22-32- is

The correct option is A 6nn+1 [ Let; ...

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Standard XII

Mathematics

Summation by Sigma Method

The sum to ...

Question

The sum to $n$ terms of the series
$\frac{3}{1^{2}} + \frac{5}{1^{2} + 2^{2}} + \frac{7}{1^{2} + 2^{2} + 3^{2}} + . . . . . . . . .$ is

\frac{6 n}{n + 1}

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\frac{9 n}{n + 1}

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\frac{12 n}{n + 1}

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\frac{3 n}{n + 1}

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Solution

The correct option is A $\frac{6 n}{n + 1}$
$\begin{matrix} L e t T_{r} = \frac{2 r + 1}{1^{2} + 2^{2} + r^{2}} = \frac{(2 r + 1)}{r (r + 1) (2 r + 1)} = \frac{6}{r (2 + 1)} \sum T_{r} = \sum \frac{6}{r (r + 1)} = 6 \sum_{r = 1}^{n} \frac{(r + 1) - r}{r (r + 1)} = 6 \sum_{r = 1}^{n} [\frac{1}{r} - \frac{1}{r + 1}] = 6 [(1 - \frac{1}{2}) + (\frac{1}{2} - \frac{1}{3}) + . . . . . . . + (\frac{1}{n} - \frac{1}{n + 1})] = 6 [1 - \frac{1}{n + 1}] = \frac{6 n}{n + 1} H e n c e, o p t i o n A i s t h e c o r r e c t a n s w e r . \end{matrix}$

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Similar questions

Q. Show that

\frac{1 \times 2^{2} + 2 \times 3^{2} + . . . + n \times {(n + 1)}^{2}}{1^{2} \times 2 + 2^{2} \times 3 + . . . + n^{2} \times (n + 1)} = \frac{3 n + 5}{3 n + 1}

Show that

$\frac{1 \times 2^{2} + 2 \times 3^{2} + \dots \dots + n \times (n + 1)^{2}}{1^{2} \times 2 + 2^{2} \times 3 + \dots \dots + n^{2} \times (n + 1)} = \frac{3 n + 5}{3 n + 1}$

Q. The sum of series

\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + . . . .

upto n terms is

Q. The sum to

n

terms of the series

\frac{3}{1^{2}} + \frac{5}{1^{2} + 2^{2}} + \frac{7}{1^{2} + 2^{2} + 3^{2}} + . . .

Q. The sum of

1

n

terms of the series

\frac{1^{2}}{1} + \frac{1^{2} + 2^{2}}{1 + 2} + \frac{1^{2} + 2^{2} + 3^{2}}{1 + 2 + 3} + . . . . . . . .

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The sum to n terms of the series 312+512+22+712+22+32+......... is

The correct option is A 6nn+1LetTr=2r+112+22+r2=(2r+1)r(r+1)(2r+1)=6r(2+1)∑Tr=∑6r(r+1)=6∑nr=1(r+1)−rr(r+1)=6∑nr=1[1r−1r+1]=6[(1−12)+(12−13)+.......+(1n−1n+1)]=6[1−1n+1]=6nn+1Hence,optionAisthecorrectanswer.

The sum to $n$ terms of the series
$\frac{3}{1^{2}} + \frac{5}{1^{2} + 2^{2}} + \frac{7}{1^{2} + 2^{2} + 3^{2}} + . . . . . . . . .$ is