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Arithmetic Sequence Calculator

Calculate the nth term and series sum from the first term and common difference.

Result

32.00
185.00

How it is calculated

An arithmetic sequence adds the same amount at every step. With first term a₁, common difference d and number of terms n, the tool computes the last term aₙ = a₁ + (n − 1) × d and the sum of the first n terms S = n / 2 × (a₁ + aₙ). The difference d may be negative or a decimal, which gives a falling or fractional sequence. The number of terms n must be at least 1 and any decimals are dropped (n = 6.9 is treated as 6). The first term and the difference can be any numbers. Results are rounded to two decimals. The sum formula works because pairing the first and last terms, the second and second-last and so on always gives the same total.

Worked example

Take the sequence 5, 8, 11, 14, … with first term 5, difference 3 and 10 terms. The 10th term is 5 + (10 − 1) × 3 = 5 + 27 = 32.00. The sum of the ten terms is 10 / 2 × (5 + 32) = 5 × 37 = 185.00. As a check, the ten terms are 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, and adding them up one by one also gives 185. The tool shows 32.00 as the nth term and 185.00 as the sum.

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Common mistakes

  • Using n × d instead of (n − 1) × d for the nth term. The first term has no difference added yet, so only n − 1 steps happen.
  • Mixing up the difference with the ratio. Adding 3 each time is arithmetic; multiplying by 3 each time is geometric and needs a different formula.
  • Treating the sum as the nth term, or entering n as the last value rather than as the number of terms.
  • Entering a decreasing sequence with a positive difference. For 20, 17, 14, … the difference is −3.

About this calculator

This calculator finds the nth term of an arithmetic sequence (aₙ = a₁ + (n − 1)d, where d is the common difference added at every step) and the sum of the sequence up to that term, from just the first term and the common difference. An arithmetic sequence is any list of numbers where each term differs from the last by the same fixed amount, whether that amount is positive, negative or a fraction.

Students use it to check sequence and series homework without writing out every term by hand, which becomes impractical once n gets into the hundreds or thousands. The same math underlies real patterns like a savings plan that adds a fixed amount each period, a venue's seating where each row has a fixed number more seats than the last, or any process that increases or decreases by a constant step.

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Comments (1)

  • Editorial team

    The tool gives the term and sum for a count you choose, but you can turn the formula around to test whether a value belongs to the sequence: n = (target − first term) ÷ d + 1. For 5, 8, 11 and so on, 50 gives n = 16, so it is the 16th term, while 51 gives about 16.33, so it never appears. Only whole results of n count. What kind of pattern are you modelling: savings steps, seating rows or something else?

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