[Math] Find common ratio of a geometric sequence given the sum

calculusratiosequences-and-series

I feel like I have tried everything with this problem. Here it is:
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Since this is an infinite sequence, I believe the formula for the sum is found with the equation firstTerm/(1-ratio).

I get 6x^7 as the first term. However, when I plug this into the formula above, I do not seem to be getting the correct answer. I am new, so forgive me for formatting issues. Thanks.

Best Answer

You'll need the summation to be from $0$ to $\infty$ to apply that formula:

$$\sum_{n=1}^{\infty} 6x^{7n} = 6 \sum_{n=1}^{\infty} (x^7)^n = -6 + 6 \sum_{n=0}^{\infty} (x^7)^n = -6 + \frac{6}{1-x^7} = 10,$$

so

$$1-x^7 = \frac{3}{8} \\ x = \sqrt[7]\frac{5}{8}.$$

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