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DanielleElmas [232]
3 years ago
10

Use geometric series to find the fraction of 0.8967898989

Mathematics
1 answer:
Sliva [168]3 years ago
8 0

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}

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