If Sr denotes the sum of the infinite geometric series whose first term is r and common ratio is 1-1-r- where r - N- then the value of -r-110 Sr2 isA- 385B- 659C- 384D- 505

The correct option is D 505From given condition, we get S_r = r1 1r+1 ⇒ S_r = r+1 Now, nbsp;∑_r=1^10S_r^2 = ∑_r=1^10(r+1)^2 nbsp;=2^2+3^2+…+11^2 nbsp ...

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Standard XII

Mathematics

Vn Method

If Sr denotes...

Question

If $S_{r}$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1 + r}$ , where $r \in N$ , then the value of $10 \sum r = 1 S_{r}^{2}$ is

505

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385

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384

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659

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Solution

The correct option is A $505$
From given condition, we get
$S_{r} = \frac{r}{1 - \frac{1}{r + 1}} \Rightarrow S_{r} = r + 1$

Now, $10 \sum r = 1 S_{r}^{2}$
$= 10 \sum r = 1 (r + 1)^{2} = 2^{2} + 3^{2} + \dots + 11^{2} = 1^{2} + 2^{2} + 3^{2} + \dots + 11^{2} - 1^{2} = \frac{11 (12) (23)}{6} - 1 = 505$

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Standard XII Mathematics

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If Sr denotes the sum of the infinite geometric series whose first term is r and common ratio is 11+r, where r∈N, then the value of 10∑r=1S2r is

The correct option is A 505From given condition, we get Sr=r1−1r+1⇒Sr=r+1 Now, 10∑r=1S2r =10∑r=1(r+1)2=22+32+…+112=12+22+32+…+112−12=11(12)(23)6−1=505

If $S_{r}$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1 + r}$ , where $r \in N$ , then the value of $10 \sum r = 1 S_{r}^{2}$ is