Arithmetic Series Sum Calculator

Formula

Series sum = Number of terms × (2 × First term+(Number of terms-1) × Common difference) ÷ 2

Worked example

With First term: 3, Common difference: 2, Number of terms: 10, series sum is 120.

Calculation method

The rule is Series sum = Number of terms × (2 × First term+(Number of terms-1) × Common difference) ÷ 2.

For this arithmetic series sum example, changing first term from 3 to 4 while keeping the other inputs fixed changes series sum from 120 to 130.

Reading series sum

The nth term identifies one position; a series sum accumulates several positions. Fibonacci terms follow their own recurrence from earlier terms. In the worked values on this page, first term 3, common difference 2, number of terms 10 produce series sum 120.

Practical use

Use these formulas for recurring patterns, repeated growth, and ordered numerical problems. The specific task here is to find series sum from first term, common difference, number of terms.

Limits and common mistakes

An infinite geometric series has a finite sum only when the absolute common ratio is below one. Check whether the requested index starts at zero or one. Verify first term before interpreting series sum; a value measured on another basis would answer a different question.

Why the relationship works

An arithmetic pattern repeatedly adds a fixed difference, while a geometric pattern repeatedly multiplies by a fixed ratio. A single nth term and a sum of the first n terms are different results. The infinite geometric sum is a separate limiting calculation that is valid only when the ratio’s absolute value is below one. The arithmetic series sum rule uses first term, common difference, number of terms in Series sum = Number of terms × (2 × First term+(Number of terms-1) × Common difference) ÷ 2; read each symbol as the field named on this page.

Interpreting another set of values

For arithmetic series sum, the fields labeled first term, common difference, number of terms are the known quantities. The calculator solves for series sum using Series sum = Number of terms × (2 × First term+(Number of terms-1) × Common difference) ÷ 2. Clarify whether the output is an integer count, a selected term, or a sum of terms. The index and the rule about order or repetition determine which mathematical object is being counted. When comparing two calculations, keep their units and input definitions identical.

For a second arithmetic series sum check, keep the other default inputs fixed and change first term from 3 to 5. Series sum changes from 120 to 140. This is a comparison of two particular cases. Repeat it near the values relevant to your task; do not assume that every formula changes at a constant rate.

Checks and limits

Check whether an index starts at zero or one and whether the result is a single term or a sum. The infinite geometric sum needs an absolute common ratio below one. A positive finite series count must be a whole number. The reported series sum is 120 in the worked example. Keep its stated unit or counting convention when checking it against another calculation.