Mathematics: Use geometric series to find the fraction of 0.8967898989

Geometric mean is the square root of the product of the two numbers:...

Use geometric series to find the fraction of 0.8967898989

crystal271

14.01.2022

I'm guessing the repeating part is 89 at the end, so that

$x=0.8967\overline89\implies10^4x=8967.\overline{89}$

Then

$10^4x=8967+\displaystyle89\sum_{i=1}^\infty\frac1{100^i}$

$10^4x=8967+89\left(\dfrac1{1-\frac1{100}}-1\right)$

$10^4x=8967+\dfrac{89}{99}$

$x=\dfrac{8967}{10^4}+\dfrac{89}{99\cdot10^4}$

$x=\dfrac{443911}{495000}$

###

An arguably quicker way without using geometric series:

$10^4x=8967.\overline{89}$

$10^6x=896789.\overline{89}$

10^6x-10^4x=887822

$x=\dfrac{887822}{10^6-10^4}=\dfrac{443911}{495000}$

0,0(0 оценок)