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  • Please,please help me - Sequence and series - JEE Main-3

The sum to infinity of the series

1+\frac{2}{3}+\frac{6}{3^{2}}+\frac{10}{3^{3}}+\frac{14}{3^{4}}+.......is

  • Option 1)

    3

  • Option 2)

    4

  • Option 3)

    6

  • Option 4)

    2

 

Answers (1)

best_answer

As we learnt

 

Sum of infinite terms of an AGP -

 

S_{\infty }= \frac{a}{1-r}+\frac{dr}{\left ( 1-r \right )^{2}}\\here \: \left | r \right |< 1

- wherein

a\rightarrow first term

d\rightarrow common diff. of a AP

r\rightarrow common ratio of a GP

 

 

Given S is an G.P........................(i)

So, \frac{1}{3}S= \frac{1}{3}+\frac{2}{9}+\frac{6}{27}+\frac{10}{81}+\frac{14}{243}+...\infty..............(ii)

(i)-(ii),

\frac{2}{3}S= \left (1+\frac{1}{3} \right )+\frac{4}{9}+\frac{4}{27}+\frac{4}{81}+...\infty

=\frac{4}{3} \left (1+\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...\infty \right )

=\frac{4}{3} *\frac{1}{1-1/3}= 2

\therefore S=3

 


Option 1)

3

Option 2)

4

Option 3)

6

Option 4)

2

Posted by

Himanshu

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