Question

Evaluate 0.2175757575....using the sum of an infinite geometric series

Answer 1

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =0.21+0.0075+0.000075+0.00000075+\dots}$

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =\dfrac{21}{100}+75(0.0001+0.000001+0.00000001+\dots)}$

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =\dfrac{21}{100}+75\left(\dfrac{1}{10^4}+\dfrac{1}{10^6}+\dfrac{1}{10^8}+\dots\right)}$

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =\dfrac{21}{100}+75\left(\dfrac{\dfrac{1}{10^4}}{1-\dfrac{1}{10^2}}\right)}$

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =\dfrac{21}{100}+\dfrac{75\times10^2}{10^4(10^2-1)}}$

$\displaystyle\longrightarrow\sf{0.21757575\dots\ =\dfrac{21}{100}+\dfrac{75}{99\times100}}$

$\displaystyle\longrightarrow\sf{\underline{\underline{0.21757575\dots\ =\dfrac{359}{1650}}}}$

Answer 2

Answer:

718/3300

Step-by-step explanation: