Quick answer
The partial sum of n terms is Sₙ = n/2 (a₁ + aₙ), equivalent to n/2 [2a₁ + (n - 1)d] when you know d.
Formula
- Sₙ = n/2 (a₁ + aₙ)
- Sₙ = n/2 [2a₁ + (n - 1)d]
Introduction
An arithmetic series answers a different question than a sequence: what is the total when you add the terms together? The formula saves time when n is large.
If you need the list before you sum, start with the calculator or read how to find terms first.
Vocabulary differences matter; see sequence vs series before you mix formulas on an exam.
Series vs sequence in practice
A sequence is written 3, 7, 11, 15. A series is 3 + 7 + 11 + 15.
Series questions often appear after sequence questions in the same word problem. First list, then sum.
The sum formula pairs the first and last term because the terms are evenly spaced. That pairing is why the average of endpoints times n works.
Sum formulas and when to use each
- Sₙ = n/2 (a₁ + aₙ)
- Sₙ = n/2 [2a₁ + (n - 1)d]
Use the first form when you already know the last term. Use the second when you know a₁ and d but not the last term explicitly.
Both forms are algebraically equivalent for arithmetic sequences. Pick whichever reduces steps with the givens you have.
For small n, adding terms manually is a legitimate check that builds trust in the formula result.
Step-by-step guide
- Identify n. Confirm how many terms are included in the sum.
- Find a₁ and aₙ. List terms or use aₙ = a₁ + (n - 1)d.
- Apply Sₙ. Compute n/2 times (a₁ + aₙ).
- Optional alternate form. Use the d form if the last term was not given directly.
- Check small cases. Add terms by hand when n is 4 or 5 to verify.
Worked sum example
Add the first four terms of 3, 7, 11, 15. Here a₁ = 3, a₄ = 15, n = 4.
S₄ = 4/2 × (3 + 15) = 2 × 18 = 36.
Direct addition also gives 3 + 7 + 11 + 15 = 36, so the formula aligns with brute force.