Sum of a Linear Number Sequence Calculator

How to use this linear sequence calculator

What this calculator does

This calculator evaluates a finite arithmetic sequence: a list in which every term changes by the same fixed amount. It returns the Final value, which is the last term after the selected number of periods, and the Sum, which is the total of every term from the first through the last. It also builds a period-by-period table and a current-state Excel workbook. It does not infer a pattern from an irregular list, predict a nonlinear trend, or sum an infinite series.

When to use it

Use it when a quantity rises or falls by a constant amount: a savings contribution that increases by the same amount each month, a training plan that adds a fixed number of repetitions each week, a straight-line production ramp, or a classroom arithmetic-series problem. The model is appropriate only when the difference between consecutive terms remains constant.

How to calculate

1. The calculator opens with a complete demonstration: an Initial value of 5, a Difference of 2, and 12 Periods. The results and Excel export are ready immediately.
2. Replace the three sample values with your own. Results update as you type. Use a period as the decimal separator; standard thousands grouping such as 1,250.5 is accepted.
3. Read the Final value to see the last term, then read the Sum for the total across all terms. Check the detail table when you need individual terms or running totals.
4. Select Download Excel to create a validated .xlsx workbook from the current inputs and results. Select Reset to clear the demonstration and all calculated content. Reset may disable the export until all three required fields contain valid values again.

Input guide

Initial value is the required first term, written as a plain integer or decimal, for example 5 or 1,250.5. It may be positive, zero, or negative. Raising it shifts every term and therefore increases both the final value and total by a predictable amount. Do not enter currency symbols, percent signs, decimal commas, or scientific notation.

Difference is the required constant step between consecutive terms. A value of 2 means add 2 each period; – 2 means subtract 2; 0 creates a constant sequence. Larger positive values make the last term and sum grow faster, while more negative values reduce them. A common mistake is entering the second term instead of the difference.

Periods is the required count of terms, not the number of gaps between terms. Enter a whole number from 1 through 5,000, such as 12. With one period, the final value and sum both equal the initial value. Fractions, zero, and negative counts are invalid.

Output guide

Sum is the exact finite arithmetic-series total in the same unit as the input values. It is driven by all three inputs and may be positive, zero, or negative. A zero sum can occur when positive and negative terms cancel; it does not imply that every term is zero. Final value is the nth term, calculated from the initial value plus the difference multiplied by one fewer than the period count.

The summary pills report the current number of terms, the signed step, and whether the sequence is increasing, decreasing, or constant. In the detail table, Period is the term index, Term value is the value at that index, and Running sum is the cumulative total through that row. These are exact identities from the same model used by the result cards and workbook.

Worked example

For the opening example, a = 5, d = 2, and n = 12. The final value is 5 + 2 × (12 – 1) = 27. The total is 12 ÷ 2 × (5 + 27) = 6 × 32 = 192. The last table row therefore shows period 12, term value 27, and running sum 192. This agrees with the equivalent formula Sₙ = n ÷ 2 × [2a + (n – 1)d].

Learn more

OpenStax explains the nth-term rule for arithmetic sequences, including how the common difference determines each successive term. Its companion section on arithmetic series and finite sums shows why multiplying the average of the first and last terms by the number of terms gives the same total.