How this calculator works
A geometric sequence multiplies by a fixed ratio at every step. A known indexed term can be scaled back to the first term. Two non-zero terms determine a power of the ratio; an even index gap can allow both a positive and a negative real ratio. The finite sum covers the displayed range. The infinite sum starts at term one and requires a ratio of magnitude below one, except for an all-zero sequence.
Formula
aₖ = a₁r^(k-1). r^(n-m) = aₙ/aₘ. Infinite sum = a₁/(1-r) for |r| < 1. Finite sums add only the selected rows.
Worked example
With first term 8 and ratio 1/2, the first four terms are 8, 4, 2, 1. Their sum is 15 and the infinite sum is 16. First term 3 and ratio 2 gives 3, 6, 12, 24, with finite sum 45.
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Frequently asked questions
What if the ratio is one?
Every term equals the first term. The finite sum is the number of terms times that value; a non-zero infinite sum diverges.
Can the ratio be negative?
Yes. Terms alternate signs. The infinite sum still converges when its magnitude is below one.
Why are zero known terms restricted?
Two zero terms do not determine a unique ratio, while a zero ratio erases information about earlier terms. Enter the first term directly for a zero-ratio sequence.