Partial Sum Calculator

A partial sum is the sum of the first n terms of an infinite (or finite) series. For example, the 5th partial sum S₅ adds only the first 5 terms together, giving you a snapshot of how the series accumulates up to that point.

For an arithmetic series with first term a₁, common difference d, and n terms, the partial sum is Sₙ = n/2 × (2a₁ + (n−1)d). Equivalently, it equals n times the average of the first and last terms: Sₙ = n × (a₁ + aₙ) / 2.

For a geometric series with first term a₁ and common ratio r, the partial sum of n terms is Sₙ = a₁ × (1 − rⁿ) / (1 − r) when r ≠ 1. If r = 1, all terms are equal to a₁ and Sₙ = n × a₁.

In an arithmetic series, each term is obtained by adding a fixed common difference (d) to the previous term. In a geometric series, each term is obtained by multiplying the previous term by a fixed common ratio (r). Their partial sum formulas are therefore different.

Yes. You can set the start index (n₀) to any non-negative integer. The calculator will generate terms beginning at that index and sum the specified number of terms from that starting point.

When r = 1, every term equals the first term a₁, so the partial sum is simply n × a₁. The standard geometric formula would involve division by zero (1 − r = 0), so this special case is handled separately.

An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). For partial sums with a finite number of terms, convergence is not a concern — the sum is always a finite number regardless of r.

This calculator supports up to 500 terms for the partial sum. For very large term counts the table is paginated so you can still inspect individual values without performance issues.