A geometric-sum formula is easier to use correctly when you can derive it. Multiplying the sum by its common ratio creates matching terms that cancel after subtraction.
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Align the original and multiplied sums
Let S = 2 + 6 + 18 + 54. Then 3S = 6 + 18 + 54 + 162. Subtract the first line from the second: 2S = 162 - 2 = 160.
Thus S = 80. The cancellation works because each term after the first is three times its predecessor.
Track the number of terms
For n terms starting at 2 with common ratio 3, the last term is 2 × 3n-1. After multiplying by 3, the new final term is 2 × 3n.
Subtraction gives 2S = 2 × 3n - 2, so S = 3n - 1. At n = 4, this is 81 - 1 = 80.
Check assumptions before generalising
A ratio of 1 needs separate handling in the usual geometric-sum formula because its denominator becomes zero. A changing ratio would require a different method entirely.
Always write the first term and the number of terms, not just the final exponent. This original explanation supports AMC 12 algebra preparation and is not an official problem or a prediction of an upcoming paper.
Continue reading: AMC 12 Trigonometry: Solve Equations Without Losing Solutions.
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Official information and further reading: Source 1
