Arithmetic Series Calculator

The sum of an arithmetic series with n terms is given by Sₙ = n/2 × [2a + (n−1)d], where a is the first term, d is the common difference, and n is the number of terms. Alternatively, it can be written as Sₙ = n/2 × (a + aₙ), where aₙ is the last term.

An arithmetic sequence is the ordered list of terms (e.g. 1, 3, 5, 7, 9), while an arithmetic series is the sum of those terms (e.g. 1 + 3 + 5 + 7 + 9 = 25). This calculator finds the sum, making it an arithmetic series calculator.

The common difference (d) is the fixed amount added to each term to get the next one. For example, in the series 2, 5, 8, 11, the common difference is 3. It can be positive, negative, or zero.

The nth term (aₙ) of an arithmetic series is calculated using aₙ = a + (n−1) × d, where a is the first term, d is the common difference, and n is the term position. This is also the last term when you are summing n terms total.

The sum of the first N natural numbers (1 + 2 + 3 + ... + N) can be found using this calculator by setting the first term to 1, the common difference to 1, and the number of terms to N. The formula gives Sₙ = N × (N + 1) / 2.

Yes, the common difference can be negative, which means the series is decreasing. For example, 20, 15, 10, 5, 0 has a common difference of −5. The sum formula works exactly the same way regardless of the sign of d.

In an arithmetic series, each term differs from the previous by a constant amount (common difference). In a geometric series, each term is multiplied by a constant ratio. For example, 1, 3, 5, 7 is arithmetic (d=2), while 1, 3, 9, 27 is geometric (r=3).

To manually sum an arithmetic series, identify the first term (a), common difference (d), and number of terms (n). Then apply Sₙ = n/2 × [2a + (n−1)d]. For instance, for 1 + 3 + 5 + ... + 19 (10 odd numbers): S = 10/2 × [2×1 + (10−1)×2] = 5 × 20 = 100.