How a geometric sequence is calculated
Each term is the previous term multiplied by a fixed common ratio r. The nth term follows directly from a₁ and r; the sum of the first n terms has a closed form that avoids adding every term one by one.
- a₁ — the first term
- r — the common ratio between consecutive terms
- n — how many terms (1st through nth)
- aₙ — the nth term
- Sₙ — the sum of the first n terms
More detail
Why r = 1 needs its own formula
The sum formula divides by (r − 1), which is undefined at r = 1. But r = 1 just means every term equals a₁, so the sum of n identical terms is simply a₁ × n — no separate derivation needed.
Negative and fractional ratios
A negative ratio like r = −2 makes the sequence alternate in sign (1, −2, 4, −8, …), while a fractional ratio like r = 0.5 makes it shrink toward zero (8, 4, 2, 1, …). The same two formulas above handle both cases without changes.
Source
Source: Standard algebra. The general term and sum formulas for geometric sequences are standard results in algebra. The nth term aₙ = a₁×r^(n−1) follows directly from the definition: starting from the first term a₁ and repeatedly multiplying by the common ratio r. The sum formula Sₙ = a₁(rⁿ−1)/(r−1) is obtained by the standard derivation of letting S be the partial sum, subtracting rS, and simplifying — a widely used technique that extends to the theory of infinite series in calculus.
Quick check. For a₁ = 1 and r = 2, the sequence is 1, 2, 4, 8, 16 — each term double the last. Notice the newest term (16) already exceeds the sum of all earlier ones (1+2+4+8 = 15): when r ≥ 2, the final term dominates the entire sum.
Frequently asked questions
What is the 5th term of 1, 2, 4, 8, …?
This sequence has a₁ = 1 and r = 2, so a₅ = 1 × 2⁴ = 16. Enter a₁ = 1, r = 2, n = 5 above to see it, along with the sum of the first 5 terms (S₅ = 31).
How do you sum a geometric sequence?
Use Sₙ = a₁(rⁿ − 1)/(r − 1) when r ≠ 1, or Sₙ = a₁ × n when r = 1 (every term is identical). This calculator applies whichever form matches your r automatically.
What happens with a negative common ratio?
The terms alternate sign — for a₁ = 1, r = −2, n = 4 the sequence is 1, −2, 4, −8, giving a₄ = −8 and S₄ = −5. Both formulas above work unchanged for negative r.
Why can't the first term or common ratio be 0?
A geometric sequence is defined by a constant ratio between consecutive nonzero terms — if a₁ = 0 every term is 0 with no meaningful ratio, and if r = 0 the sequence collapses after the first term. Neither is a valid geometric sequence, so this calculator shows “—” for those inputs.