Answer:
75/99.
Step-by-step explanation:
The infinite geometric sum formula when \(r<1\) is
\(\sum ar^{n}=\frac{a}{1-r}\)
So, we know that
\(a=\frac{75}{100}\), because it represents the beginning of the number.
\(r=\frac{1}{100}\), because the periodic number repeats each two digits.
Replacing these values, we have
\(\sum (\frac{75}{100} )(\frac{1}{100})^{n}=\frac{\frac{75}{100} }{1-\frac{1}{100}}\\\frac{\frac{75}{100}}{\frac{100-1}{100} }=\frac{\frac{75}{100} }{\frac{99}{100} }=\frac{75}{99}=0.75757575...\)
Therefore, the fraction would be 75/99.
If you wanna demonstrate this is true, you just have to divide, then the quotient will be a rational infinite number like the given one 0.757575...
