If the sum of the first 5 terms of a geometric series is 31 and the first term i

If the sum of the first 5 terms of a geometric series is 31 and the first term is 1, what is the common ratio? (2023)

If the sum of the first 5 terms of a geometric series is 31 and the first term is 1, what is the common ratio? (2023)

Step 1: Identify the first term (a) of the geometric series. Here, a = 1.
Step 2: Identify the number of terms (n) we are summing. Here, n = 5.
Step 3: Write down the formula for the sum of the first n terms of a geometric series: S_n = a(1 - r^n) / (1 - r).
Step 4: Substitute the known values into the formula: S_5 = 1(1 - r^5) / (1 - r).
Step 5: Set the equation equal to the given sum, which is 31: 1(1 - r^5) / (1 - r) = 31.
Step 6: Simplify the equation: (1 - r^5) / (1 - r) = 31.
Step 7: Multiply both sides by (1 - r) to eliminate the denominator: 1 - r^5 = 31(1 - r).
Step 8: Distribute 31 on the right side: 1 - r^5 = 31 - 31r.
Step 9: Rearrange the equation to isolate r: r^5 - 31r + 30 = 0.
Step 10: Solve the equation for r. By testing possible values, we find that r = 3 is a solution.

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