Arithmetic Series Sum
Arithmetic Series Sum
Arithmetic sequences — quantities that increase or decrease by a fixed amount each step — appear whenever something accumulates linearly: equal loan payments, evenly spaced structural bolt loads, or a production line ramping output by a constant amount each shift. Summing many such terms by hand would be tedious, but a closed-form formula avoids adding them one at a time.The formula pairs the first and last terms, the second and second-to-last, and so on — each pair sums to the same value, which is the trick behind the classic "Gauss summed 1 to 100 in his head" story and generalizes directly to any starting value and step size.
The sum of the first n terms of an arithmetic series is S = (n/2)(2a_1 + (n−1)d). where a_1s is the first term, d_s is the common difference between consecutive terms, n_s is the number of terms, and S_arith is the total sum.
Inputs
a_1s := 5=5
d_s := 3=3
n_s := 20=20
Combine the first term, the step size, and the term count using the closed-form sum — this avoids adding all twenty terms one by one.
S_arith := n_s/2·(2·a_1s+(n_s-1)·d_s)=670
Results
Summing 20 terms starting at 5 and increasing by 3 each time gives a total in the several-hundreds range, matching the intuition that the average term (roughly the midpoint value) times the number of terms should be in that ballpark. This formula scales to any number of terms without ever looping term by term, which matters when n grows into the thousands.
Assumptions & limitations
- Constant common difference between consecutive terms.
- n is a positive whole number of terms.
- First term and difference are known exactly.