Nth term and sum of a geometric series.
aₙ = a · rⁿ⁻¹ Sₙ = a · (1 − rⁿ) / (1 − r), for r ≠ 1 Sₙ = n · a, for r = 1 S∞ = a / (1 − r), for |r| < 1
A geometric sequence is fully defined by its first term (a) and its common ratio (r). Three core formulas power this calculator:
1. nth Term: The value of any term in the sequence.
2. Sum of First n Terms (Partial Sum): The total of the first n terms. This formula differs depending on whether r equals 1.
3. Infinite Sum (when |r| < 1): When the absolute value of the common ratio is less than 1, the infinite series converges to a finite value.
nth term: a₆ = 3 · 2^(6−1) = 3 · 2⁵ = 3 · 32 = 96 Partial sum: S₆ = 3 · (1 − 2⁶) / (1 − 2) = 3 · (1 − 64) / (−1) = 3 · (−63) / (−1) = 3 · 63 = 189 Sequence: 3, 6, 12, 24, 48, 96
Result: 6th term (a₆) = 96 | Sum of first 6 terms (S₆) = 189 | Infinite sum: not applicable (r = 2, |r| ≥ 1)
With a first term of 3 and a common ratio of 2, every term doubles the previous one. The 6th term reaches 96, and the running total of all six terms is 189. Because the ratio r = 2 is greater than 1 in absolute value, the sequence grows without bound and has no finite infinite sum — the series diverges.
A sequence is geometric if the ratio between consecutive terms is constant. For example, 5, 15, 45, 135 … has a common ratio of 3 because each term is three times the previous one. Contrast this with an arithmetic sequence, where the difference between terms is constant.
The infinite geometric series converges only when |r| < 1. When |r| ≥ 1, adding infinitely many terms produces an infinitely large (or oscillating) result. The classic example is the series 1 + 1/2 + 1/4 + 1/8 + … = 2, where a = 1 and r = 1/2.
| Feature | Geometric | Arithmetic | |---|---|---| | Pattern | Multiply by r | Add d | | nth Term | a · rⁿ⁻¹ | a + (n−1)d | | Sum | a(1−rⁿ)/(1−r) | n/2 · (2a + (n−1)d) | | Infinite Sum | a/(1−r) if |r|<1 | Diverges always |
A **geometric sequence** is the ordered list of terms (e.g., 2, 6, 18, 54). A **geometric series** is the *sum* of those terms (e.g., 2 + 6 + 18 + 54 = 80). This calculator handles both: it lists sequence terms and computes the partial or infinite series sum.
Use the direct formula aₙ = a · rⁿ⁻¹. You only need the first term, the common ratio, and the desired position n. There is no need to compute every intermediate term.
A geometric series diverges (grows without limit or oscillates infinitely) whenever |r| ≥ 1. This means r ≥ 1 or r ≤ −1. Only when −1 < r < 1 does the infinite series converge to a finite value.
No. If a = 0, every subsequent term is also 0, and the common ratio is undefined (0/0). A geometric sequence requires a non-zero first term.
The nth term formula aₙ = a · rⁿ⁻¹ is essentially a discrete exponential function. Continuous exponential growth f(x) = a · eˣ is the continuous analogue. This is why geometric sequences model phenomena like compound interest and radioactive decay that are often described with exponential equations.
The common ratio (r) is the fixed multiplier between consecutive terms. You can find it by dividing any term by the term that immediately precedes it: r = aₙ / aₙ₋₁. For example, in the sequence 4, 12, 36, 108, the ratio is 12/4 = 3.
Yes. A negative common ratio causes the terms to alternate in sign. For instance, with a = 5 and r = −2, the sequence is 5, −10, 20, −40, 80 … The partial sum formula still works correctly with negative values of r.
Use the nth term formula and solve for n: aₙ = a · rⁿ⁻¹ → n = log(aₙ/a) / log(r) + 1. This requires a, r, and the last known term aₙ.
When |r| < 1, each additional term contributes less and less to the total. The partial sum approaches — but never exceeds — the infinite sum S∞ = a/(1−r). For a = 1 and r = 1/2, the infinite sum equals 2.
Absolutely. Geometric series underpin compound interest and annuity formulas. However, for precise financial decisions, always consult a licensed financial advisor, as real-world calculations may include fees, taxes, and rounding conventions not captured here.
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