Sum a Finite Arithmetic Sequence

Confirm that the difference is constant

A sequence sum calculator needs to know what kind of sequence it receives. This page handles a finite arithmetic sequence: each term after the first differs from the previous term by the same common difference. Enter the first term, common difference, and positive integer number of terms. The result shows the last term and sum through two equivalent formulas.

It does not infer a pattern from a pasted list. Pattern inference can be ambiguous because a finite set of observed values can fit many rules. The visitor states the arithmetic rule directly.

Last-term formula

Starting at a1, reaching the nth term requires n−1 steps. Each step adds d:

an = a1 + (n − 1)d

If a1=5, d=3, and n=8, the last term is 5+7×3=26. The bounded preview is 5, 8, 11, 14, 17, 20, 23, 26. For a large term count, the page shows only the first and last five terms with an ellipsis; it never generates a million-item DOM list.

A positive difference makes the sequence increasing, a negative difference makes it decreasing, and zero creates a constant sequence. These descriptions refer only to term order, not to growth in a real system.

Pairing formula for the sum

The first and last terms add to the same total as the second and next-to-last terms. Averaging first and last and multiplying by the count gives:

Sn = n/2 × (a1 + an)

For the example, each paired average is (5+26)/2=15.5. Eight terms produce 8×15.5=124. The result is finite because n is supplied.

Substituting the last-term expression produces a second formula:

Sn = n/2 × (2a1 + (n − 1)d)

With the same inputs, 8/2×(10+21)=4×31=124. The tool calculates both paths and reports a cross-check difference. A small difference can arise for decimal inputs under floating-point arithmetic; a large difference indicates a calculation error or overflow.

Decreasing sequence example

Enter a1=20, d=−2, and n=6. The terms are 20, 18, 16, 14, 12, and 10. The last term formula gives 20+5×(−2)=10. The sum is 6/2×(20+10)=90.

The negative common difference does not make the sum negative because all six terms remain positive. Extending the same sequence far enough can cross zero; the formula continues to add the stated terms. Interpretation depends on what they represent.

One term and constant sequences

For n=1, there are zero steps. The last term equals the first, and the sum equals that single term. The formula gives 1/2×(2a1)=a1. This boundary case should not require a special invented second term.

For d=0, every term equals a1. A sequence with a1=7 and n=12 has sum 84. The direction label says constant. These cases are useful tests for off-by-one errors.

Decimal terms and finite precision

Arithmetic sequences can use decimal first terms and differences. Enter 0.1, 0.1, and 10. The mathematical last term is 1 and the sum is 5.5. Binary floating point may produce an intermediate such as 0.9999999999999999. The result formats after calculation and shows the cross-check.

The page does not claim exact decimal arithmetic or arbitrary precision. For integer inputs within safe magnitude, the values are often exact in browser arithmetic; sufficiently large integers can exceed exact-number range. Nonfinite results are rejected rather than displayed as ordinary answers.

Arithmetic versus geometric sequence

An arithmetic sequence adds a constant difference. A geometric sequence multiplies by a constant ratio. The terms 3, 6, 12, 24 are geometric with ratio two, not arithmetic, because their differences change. This page does not apply the arithmetic formula to such a list.

The exponent calculator can evaluate a single power, but the catalog has no geometric-series tool. Use a method whose rule is explicitly defined if that sum is needed.

Interpretation limits

A finite series formula does not predict that a real trend continues. Sales, populations, savings, performance, and physical measurements rarely change by a perfectly constant difference without uncertainty. The tool does not fit data, forecast, estimate probability, or attach units.

If terms carry currency, length, time, or another unit, the sum uses that unit. The common difference must use the same unit as the terms. Calendar schedules and payment streams can require irregular dates, interest, or rules absent from this arithmetic model.

Frequently asked questions

Enter the first term, common difference, and positive term count, then compare both displayed sum formulas before copying the result.