A series is the sum of the terms in a sequence, and two of the most important types appear throughout mathematics, finance, and the physical sciences. An arithmetic series adds up terms that increase by a fixed amount each step, called the common difference, like 1 + 3 + 5 + 7. A geometric series multiplies each term by a fixed ratio to get the next one, like 1 + 0.5 + 0.25 + 0.125. Geometric series appear in compound interest calculations, radioactive decay, signal processing, and probability. Arithmetic series appear in problems involving equally spaced values, such as the total of the first n positive integers or the cost of items priced at equal increments. This calculator handles both types. Select the series type, enter the first term a, the common ratio r (for geometric) or common difference d (for arithmetic), and the number of terms n. The calculator returns the partial sum, the value of the last term, the infinite sum if the geometric series converges, and the formula used. The geometric default uses a = 1, r = 0.5, n = 10 giving a sum of 1.9980. The arithmetic default uses a = 1, d = 2, n = 10 giving a sum of 100. Switch between the two types using the selector below.
1.9980
partial sum (Sn)
Last term (an)0.0020
Infinite sum2.0000
Formulaa(1-r^n)/(1-r)
Infinite sum is shown only when |r| < 1 for geometric series. Arithmetic series have no finite infinite sum.
How it works
For a geometric series with first term a, common ratio r, and n terms: the partial sum is a * (1 - r^n) / (1 - r) when r is not 1, or a * n when r = 1. The last term is a * r^(n-1). The infinite sum a / (1 - r) exists only when |r| is strictly less than 1. For an arithmetic series with first term a, common difference d, and n terms: the partial sum is n * (2a + (n-1) * d) / 2. The last term is a + (n-1) * d. There is no finite infinite sum for an arithmetic series.
Worked example
Geometric series: a = 1, r = 0.5, n = 10. Sum = 1 * (1 - 0.5^10) / (1 - 0.5) = 0.9990234375 / 0.5 = 1.9980. Last term = 1 * 0.5^9 = 0.0020. Infinite sum = 1 / (1 - 0.5) = 2.0000. Arithmetic series (switch type): a = 1, d = 2, n = 10. Sum = 10 * (2 + 9 * 2) / 2 = 10 * 20 / 2 = 100. Last term = 1 + 9 * 2 = 19.
