Formulas
If r = 1, every term equals a₁, so Sₙ = n · a₁ — that case is handled separately by the calculator.
Worked example
Sequence 2, 6, 18, 54, ... with a₁ = 2 and r = 3. Finding the 10th term and S₁₀:
- a₁₀ = 2 · 3⁹ = 2 · 19683 = 39 366
- S₁₀ = 2 · (1 − 3¹⁰) / (1 − 3) = 2 · (−59048) / (−2) = 59 048
Because |r| = 3 > 1, the infinite sum does not converge — partial sums grow forever.
Convergence example
Take a₁ = 1, r = ½. Then S∞ = 1 / (1 − ½) = 2, even though we add infinitely many terms: 1 + ½ + ¼ + ⅛ + ... = 2. The chart's log scale is useful for the divergent case (large |r|), where successive terms span many orders of magnitude.
FAQ
Can the ratio be negative?
Yes. With r < 0 the terms alternate in sign (e.g., a₁ = 4, r = −0.5 gives 4, −2, 1, −0.5, ...). The infinite sum still converges when |r| < 1.
What happens when r = 1?
Every term equals a₁, so the sequence is constant and Sₙ = n · a₁. The standard sum formula divides by (1 − r) which is zero, so the calculator falls back to this special case automatically.