If the sum of the first n terms of a geometric series is 81, and the first term is 3, what is the common ratio?

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Q: If the sum of the first n terms of a geometric series is 81, and the first term is 3, what is the common ratio?

Solution: Using the formula S_n = a(1 - r^n) / (1 - r), we have 81 = 3(1 - r^n) / (1 - r). Solving gives r = 3.

Steps: 8

Step 1: Identify the formula for the sum of the first n terms of a geometric series, which is S_n = a(1 - r^n) / (1 - r).
Step 2: Substitute the known values into the formula. Here, S_n = 81 and a = 3, so we have 81 = 3(1 - r^n) / (1 - r).
Step 3: Multiply both sides of the equation by (1 - r) to eliminate the fraction: 81(1 - r) = 3(1 - r^n).
Step 4: Distribute 81 on the left side: 81 - 81r = 3(1 - r^n).
Step 5: Expand the right side: 81 - 81r = 3 - 3r^n.
Step 6: Rearrange the equation to isolate terms involving r: 81 - 3 = 81r - 3r^n, which simplifies to 78 = 81r - 3r^n.
Step 7: Rearrange it further to get a standard form: 3r^n - 81r + 78 = 0.
Step 8: Solve for r using trial and error or substitution. Testing r = 3 gives a valid solution.