The rule
aₙ = a₁ + (n − 1)d · Sₙ = n(a₁ + aₙ) ÷ 2
Why it works
An arithmetic sequence adds the same amount each step. To reach term n you take n−1 steps from the first, and the sum has a closed form because pairing the ends always gives the same total.
How to do it by hand
- Count the steps: the term number minus one
- Multiply by the common difference
- Add the first term
- For the sum, average the first and last and multiply by how many there are
What is worth knowing
The sum formula is the same pairing argument Gauss is said to have found as a child, generalised: first plus last, second plus second-to-last, and so on all give the same total. Arithmetic sequences model anything that grows by a fixed amount — simple interest, straight-line depreciation, a savings plan with fixed deposits. The moment growth becomes proportional rather than fixed you need a geometric sequence instead, and the difference between the two is exactly the difference between simple and compound interest.