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Arithmetic Series Sum Calculator

Enter the first term, common difference and number of terms to sum the series.

Result

258.00
38.00 (term 12)
21.50

How it is calculated

An arithmetic series adds terms that grow by a fixed step. The tool takes the first term a, the common difference d (it can be negative or zero) and the number of terms n. It computes the last term an = a + (n − 1) × d, the sum S = n ÷ 2 × (2a + (n − 1) × d), and the average term S ÷ n, which is also the midpoint of the first and last terms. The sum is the same as n × (a + an) ÷ 2. The count n must be a whole number of at least 1: a decimal such as 12.5 or a value of 0 gives no result. Results are shown to two decimals, and the third output labels which term number the last term is.

Worked example

Start at a = 5, add d = 3 each time, for n = 12 terms: 5, 8, 11, and so on. Last term: 5 + 11 × 3 = 38. Sum: 12 ÷ 2 × (2 × 5 + 11 × 3) = 6 × 43 = 258, shown as 258,00 in Turkish and 258.00 in English. Average term: 258 ÷ 12 = 21.50, which matches the midpoint of 5 and 38, since (5 + 38) ÷ 2 = 21.5. Adding the twelve terms one by one would give the same 258.

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Common mistakes

  • Entering the last term where the count belongs. Field n is how many terms there are, not the final value.
  • Getting the sign of d wrong for a falling series. Counting down 20, 17, 14 needs d = −3, not 3.
  • Counting the terms wrongly. The numbers 5 to 38 in steps of 3 are 12 terms, not 11, because both ends are included.
  • Using this for a series that multiplies by a fixed factor, such as 2, 4, 8. That is a geometric series and needs a different formula.

About this calculator

An arithmetic series is the sum of a sequence where each term increases by the same fixed amount, called the common difference (d). Its sum is found without adding every term individually, using the formula S = n/2 × (2a + (n−1)d), where a is the first term and n is the number of terms — a shortcut famously attributed to a young Gauss summing 1 through 100.

This calculator takes the first term, common difference, and number of terms, and returns the total sum along with the nth (last) term and the average term (which for any arithmetic series equals the sum divided by n, and also the midpoint between the first and last terms). It's a standard tool for algebra and precalculus students checking series problems or exploring how the formula behaves for different sequences.

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Comments (1)

  • Editorial team

    A decreasing schedule works too, by giving the difference a minus sign. Eight monthly amounts starting at 100 and falling by 5 each month end at 65.00 and total 660.00, with a mean of 82.50. The step is flat, so this is not compound growth; interest or percentage increases need a geometric series instead. What sequence are you adding up, and would a geometric version be useful to you?

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