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Common Difference Calculator

Find the common difference of an arithmetic sequence given its first term, an nth term and the term position.

How it is calculated

In an arithmetic sequence every term is the previous one plus a constant d. The n-th term is an = a1 + (n − 1) × d. Solving for d gives d = (an − a1) ÷ (n − 1), which is what the tool computes. Here a1 is the first term, an is the term you know at position n, and n − 1 is the number of steps between them. The result is rounded to two decimals, so a difference of 1/3 appears as 0.33. A negative difference means the sequence decreases, and 0 means every term is equal. The position n must be a whole number greater than 1: n = 1 has no steps and would divide by zero, and a fractional position such as 2.5 has no meaning, so both show no result. The terms themselves may be any numbers, negative or decimal.

Worked example

The first term is 4 and the 7th term is 31. Steps from the first term to the 7th: 7 − 1 = 6. Total change: 31 − 4 = 27. Common difference: 27 ÷ 6 = 4.50. Check by building the sequence: 4, 8.5, 13, 17.5, 22, 26.5, 31. The seventh term is 31, so d = 4.50 is correct, and each term is 4.50 larger than the one before.

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Common mistakes

  • Dividing by n instead of n − 1. Between the first and the 7th term there are 6 steps, not 7; dividing 27 by 7 gives 3.86, which is wrong.
  • Subtracting in the wrong order. The formula is the later term minus the first term; swapping them flips the sign of the difference.
  • Entering n = 1. The first term compared with itself has no step, so there is no result.
  • Using a fractional position such as 2.5. A term position must be a whole number, and a fractional one shows no result.
  • Using the tool for a sequence that multiplies by a fixed amount. That is a geometric sequence, and its common ratio is not found by subtracting.

About this calculator

In an arithmetic sequence, each term is found by adding a fixed amount, called the common difference, to the term before it. Given any two terms and how far apart they are in the sequence, the common difference can be found by dividing the gap between the two term values by the number of steps between them.

Students and teachers use this constantly in algebra coursework, since arithmetic sequences are one of the first patterns introduced in sequence and series topics. It also shows up in scheduling and planning problems, such as figuring out a fixed weekly savings increase or a stepped pricing structure, where each stage increases by the same fixed amount as the one before it.

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