If S -1-4-3-9-32-16-33- then 2 S - is

Let S_∞=1+43+93^2+163^3+……∞⋯(i) 13S_∞= 13+43^2+93^3+163^4+……∞⋯(ii) From (i) (ii) 23S_∞=1+33+53^2+73^3+……∞⋯(iii) 23^2S_∞= 13+33^2+53^3+……∞⋯(iv) from (iii) ...

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Byju's Answer

Standard XII

Mathematics

Method of Difference

If S ∞=1+4/3+...

Question

If $S_{\infty} = 1 + \frac{4}{3} + \frac{9}{3^{2}} + \frac{16}{3^{3}} + \dots \dots$ , then $2 S_{\infty}$ is

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Solution

Let $S_{\infty} = 1 + \frac{4}{3} + \frac{9}{3^{2}} + \frac{16}{3^{3}} + \dots \dots \infty \dots (i)$
$\frac{1}{3} S_{\infty} = \frac{1}{3} + \frac{4}{3^{2}} + \frac{9}{3^{3}} + \frac{16}{3^{4}} + \dots \dots \infty \dots (i i)$
From $(i) - (i i)$
$\frac{2}{3} S_{\infty} = 1 + \frac{3}{3} + \frac{5}{3^{2}} + \frac{7}{3^{3}} + \dots \dots \infty \dots (i i i) \frac{2}{3^{2}} S_{\infty} = \frac{1}{3} + \frac{3}{3^{2}} + \frac{5}{3^{3}} + \dots \dots \infty \dots (i v)$
from $(i i i) - (i v)$
$\frac{4}{9} S_{\infty} = 1 + \frac{2}{3} + \frac{2}{3^{2}} + \frac{2}{3^{3}} + \dots \dots \infty = 1 + 2 (\frac{1 / 3}{1 - 1 / 3}) = 2 2 S_{\infty} = 9$

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Similar questions

Q. If

S_{\infty} = 1 + \frac{4}{3} + \frac{9}{3^{2}} + \frac{16}{3^{3}} + \dots \dots

, then

2 S_{\infty}

Q. If

S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{3}{3^{3}} + \frac{4}{3^{4}} + \dots \infty,

then the value of

4 S

Q. If

S = \frac{1}{3} + \frac{2}{3^{2}} + \frac{3}{3^{3}} + \frac{4}{3^{4}} + \dots \infty,

then the value of

4 S

Q. if

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

up to

\infty

, then:

1, 1, 1, 2, 4, 8, 3, 9, 27, 4, 16, 32

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If S∞=1+43+932+1633+⋯⋯, then 2S∞ is

Let S∞=1+43+932+1633+……∞⋯(i) 13S∞= 13+432+933+1634+……∞⋯(ii) From (i)−(ii) 23S∞=1+33+532+733+……∞⋯(iii)232S∞= 13+332+533+……∞⋯(iv) from (iii)−(iv) 49S∞=1+23+232+233+……∞=1+2(1/31−1/3)=22S∞=9

If $S_{\infty} = 1 + \frac{4}{3} + \frac{9}{3^{2}} + \frac{16}{3^{3}} + \dots \dots$ , then $2 S_{\infty}$ is