Question

Find value of 0.42323232323..........(QUESTION OF GP)​

Answer 1

SOLUTION:

Let the given value be some variable let say x.

$x = 0.4232323...$

Can be written as,

$x = 0. 4\overline{23}$

Can also be expressed as,

${x = 0.4 + (0.023 + 0.00023 + 0.0000023 + ...)}$

Now, we can observe that the series in the bracket is forming an infinite GP where first term is 0.023 and the common ratio is 1/100 or 0.01.

We have formula for sum of infinite GP.

$\boxed{S_\infty = \frac{a}{1 - r}}$

Note: In an infinite GP, common ratio lies between the internal (0, 1) otherwise the series will diverge.

By using the formula, we get:

${x = 0.4 + \left ( \dfrac{0.023}{1 - \frac{1}{100}} \right)}$

${x = 0.4 + \left ( \dfrac{ \frac{23}{1000} }{ \frac{99}{100}} \right)}$

${x = 0.4 +\dfrac{23}{990}}$

${x = \dfrac{4}{10} +\dfrac{23}{990}}$

${x = \dfrac{396 + 23}{990}}$

${x = \dfrac{419}{990}}$

$x = \dfrac{313}{495}$

This is the required answer.