For students and parents · O-Level A-Math
Arithmetic and Geometric Progressions, Worked Cleanly
Sequences built by adding, or by multiplying.
Mrs Eileen Toh
Founder & Curriculum Architect · ex-MOE · 3 min read · Updated 29 Jun 2026
A progression is a sequence built by a fixed rule. In an arithmetic progression you add the same number each step; in a geometric progression you multiply by the same number each step. The two have parallel formulas, one for the term you want and one for the sum of a run of terms, and the first job in any question is to decide which kind you are looking at.
Arithmetic progressions
You add a common difference d each step, starting from a first term a.
The nth term: T_n = a + (n - 1)d.
The sum of the first n terms: S_n = (n/2)(2a + (n - 1)d).
Worked example. For the progression with a = 3 and d = 4 (so 3, 7, 11, 15, ...): the 10th term is T_10 = 3 + 9 x 4 = 39, and the sum of the first 10 terms is S_10 = (10/2)(2 x 3 + 9 x 4) = 5 x 42 = 210.
Geometric progressions
You multiply by a common ratio r each step.
The nth term: T_n = a rn-1.
The sum of the first n terms: S_n = a(rn - 1)/(r - 1).
Worked example. For a = 2 and r = 3 (so 2, 6, 18, 54, ...): the 5th term is T_5 = 2 x 34 = 162, and the sum of the first 5 terms is S_5 = 2(35 - 1)/(3 - 1) = 2 x 242/2 = 242.
The sum to infinity
A geometric progression has a finite total even with infinitely many terms, but only when the ratio is between minus 1 and 1, so the terms shrink. Then:
S_infinity = a/(1 - r).
Worked example. For a = 8 and r = 1/2 (so 8, 4, 2, 1, ...): S_infinity = 8/(1 - 0.5) = 16. The condition |r| < 1 is essential; quote it whenever you use this formula, because the marks include the condition, not just the number.
The teaching point
The two progressions run in parallel: arithmetic adds and uses 2a + (n - 1)d, geometric multiplies and uses rn, and only the geometric one, with a ratio smaller than 1 in size, has a sum to infinity. The first decision is which kind you have; the second is which formula the question wants, the term or the sum. Most lost marks are a mix-up between the two progressions or omitting the |r| < 1 condition.
If your child can quote the formulas but mixes up arithmetic and geometric, or forgets the condition on the sum to infinity, that is a clear, teachable gap. GPA's Secondary Math programme drills them apart on real A-Math papers. A short diagnostic consult will show where the confusion sits.
Every term and sum on this page was checked independently before publishing.