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How would I go about finding a variable in the summation equation if I am given the answer to the summation as so: $\sum \limits_{n=1}^{\infty} 7(x)^{n-1} =25$

I know that x has to be less than 1 because anything above 1 will simply diverge. I know that the first term in the series will always be 1 when n = 1. However I cannot seem to figure out how to narrow it down any further. How would I solve for x here?

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1 Answer

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Okay so based off of the hints I've got, I think I figured out how to answer this now. Hopefully this will help others who might see this in the future.

So I start with $\sum \limits_{n=1}^{\infty} 7(x)^{n-1} =25$

I pull the 7 to the front and get $7\sum \limits_{n=1}^{\infty} (x)^{n-1} =25$

I divide both sides by 7 to get rid of it and get $\sum \limits_{n=1}^{\infty} (x)^{n-1} =\frac{25}{7}$

Using the formula for geometric series I can substitute $\sum \limits_{n=1}^{\infty} (x)^{n-1}$ for $\frac{1}{1-x}$ leaving me with

$\frac{1}{1-x} = \frac{25}{7}$

Solving for that gives me x = $\frac{18}{25}$ which was the correct answer.

Thanks to everyone for the help! If I got anything wrong please feel free to correct any of my errors.

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