A worked example: the 12th term of 5, 8, 11, ...
Set the Sequence Calculator to arithmetic with a first term of 5, a common difference of 3, and n = 12, and it returns an nth term of 38 and a sum of the first n terms of 258. Both come straight from the closed-form formulas: the nth term is a₁ + (n − 1)d, which here is 5 + (11 × 3) = 5 + 33 = 38. The sum is n ÷ 2 × (2a₁ + (n − 1)d), which is 6 × (10 + 33) = 6 × 43 = 258.
The formula: nth term and sum, in closed form
For an arithmetic sequence, the nth term is a₁ + (n − 1)d and the sum of the first n terms is n ÷ 2 × (2a₁ + (n − 1)d). For a geometric sequence, the nth term is a₁ × r^(n − 1), and the sum of the first n terms is a₁ × (rⁿ − 1) ÷ (r − 1) — or simply a₁ × n in the special case where r = 1. These formulas compute any term directly, without building the whole list first. Generating the same arithmetic sequence (a₁ = 5, d = 3) term by term for n = 1 through 6 gives 5, 8, 11, 14, 17, 20 — and asking the calculator directly for n = 6 returns exactly 20, confirming the closed form matches straightforward repeated addition.
A geometric example: halving 1,000 eight times
Set the calculator to geometric with a first term of 1,000, a common ratio of 0.5, and n = 8, and it returns an nth term of 7.8125 and a sum of the first 8 terms of 1,992.1875. Listing all eight terms shows the halving directly: 1000, 500, 250, 125, 62.5, 31.25, 15.625, 7.8125. Because the ratio's absolute value is less than 1, each new term contributes less than the last, and the running sum climbs toward — but never quite reaches — 2,000 (twice the first term) as more terms are added.
Why closed-form beats adding term by term
The calculator accepts any term number up to 1,000. Computing the 1,000th term by repeated addition or multiplication would mean carrying out 999 separate steps; the closed-form formulas get there in one calculation regardless of how large n is. That matters beyond textbook exercises — the same arithmetic and geometric formulas underpin things like projecting a fixed monthly increment far into the future, or modeling repeated percentage decay, where you often care about a specific far-off term or a total sum rather than every value in between.