Question

find sum of infinity
3, 6/5 , 9/25 , 12/125 , 15/625, .......​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{3,\dfrac{6}{5},\dfrac{9}{25},\dfrac{12}{125},........}$

$\underline{\textsf{To find:}}$

$\textsf{Sum to infinite terms of the given sequence}$

$\underline{\textsf{Solution:}}$

$\textsf{Consider,}$

$\mathsf{3,\dfrac{6}{5},\dfrac{9}{25},\dfrac{12}{125},........}$

$\textsf{clearly, it is an arithmetico-geometric sequence}$

$\mathsf{with\;a=3,r=\dfrac{1}{5}\;and\;d=3}$

$\textsf{Then,}$

$\textsf{Sum to infinite terms of the sequence}$

$\mathsf{S_{\infty}=\dfrac{a}{1-r}+\dfrac{dr}{(1-r)^2}}$

$\mathsf{S_{\infty}=\dfrac{1}{1-\dfrac{1}{5}}+\dfrac{3(\dfrac{1}{5})}{(1-\dfrac{1}{5})^2}}$

$\mathsf{S_{\infty}=\dfrac{1}{\dfrac{4}{5}}+\dfrac{\dfrac{3}{5}}{(\dfrac{4}{5})^2}}$

$\mathsf{S_{\infty}=\dfrac{5}{4}+\dfrac{\dfrac{3}{5}}{\dfrac{16}{25}}}$

$\mathsf{S_{\infty}=\dfrac{5}{4}+\dfrac{3}{5}{\times}\dfrac{25}{16}}$

$\mathsf{S_{\infty}=\dfrac{5}{4}+\dfrac{3}{1}{\times}\dfrac{5}{16}}$

$\mathsf{S_{\infty}=\dfrac{5}{4}+\dfrac{15}{16}}$

$\mathsf{S_{\infty}=\dfrac{20+15}{16}}$

$\mathsf{S_{\infty}=\dfrac{35}{16}}$

$\implies\boxed{\mathsf{S_{\infty}=\dfrac{35}{16}}}$

$\underline{\textsf{Answer:}}$

$\textsf{The sum to infinite terms of the given sequence is}\;\mathsf{\dfrac{35}{16}}$

Find more:

Find the sum to infinite terms of the series 1+5x+9x^2+13x^3+.....

Where |x| <1

https://brainly.in/question/25340075

Answer 2

Answer:

Step-by-step explanation:

45/4