Summation Calculator
Calculate the sum of an arithmetic or geometric series in one step. Enter the first term, common difference (or ratio), and the number of terms. For example, the arithmetic series 2 + 5 + 8 + 11 + 14 (first term 2, difference 3, 5 terms) sums to 40.
Arithmetic Series
In an arithmetic series, each term increases by a constant difference d. The sum of n terms is:
S = n/2 x (first + last) or equivalently S = n/2 x (2a + (n-1)d)
Example: Sum of 1 + 2 + 3 + ... + 100 = 100/2 x (1 + 100) = 5,050. This is the famous result Gauss reportedly discovered as a child.
Geometric Series
In a geometric series, each term is multiplied by a constant ratio r. The sum of n terms is:
S = a(1 - r^n) / (1 - r) (when r is not equal to 1)
Example: 3 + 6 + 12 + 24 + 48 (first term 3, ratio 2, 5 terms) = 3(1 - 32)/(1 - 2) = 3 x 31 = 93.
When to Use Each
- Arithmetic: equally spaced numbers (1, 4, 7, 10, ...) or uniform increments
- Geometric: doubling, halving, exponential growth, compound interest, population models
Infinite Geometric Series
If the absolute value of r is less than 1, the infinite geometric series converges: S = a / (1 - r). For example, 1 + 1/2 + 1/4 + 1/8 + ... = 1 / (1 - 0.5) = 2.