SL 1.2—Arithmetic sequences and series

An arithmetic sequence changes by the same common difference d at every step. If the first term is a, the nth term is uₙ = a + (n − 1)d; the index n counts terms, so the first term uses n = 1.

For 7, 11, 15, …, d = 4. The 20th term is 7 + 19(4) = 83, and the sum of the first 20 terms is 20(7 + 83)/2 = 900. The same sum formula pairs the first and last terms, which is why the average term is (a + uₙ)/2.

A sequence lists terms; a series adds them. Check that the claimed common difference is constant before using an arithmetic formula, and do not replace n − 1 with n when finding a term.

Sigma form makes the finite series explicit: $S_n=\sum_{k=1}^{n}[a+(k-1)d]$. In a simple-interest model, equal interest added each period produces an arithmetic sequence. For principal $1000$ at 4% simple interest, the yearly balances are $1040,1080,1120,\ldots$ with common difference $40$; real data may require an approximate common difference rather than a perfect one.