Progression in Maths - AP, GP, HP, Formulas

TL;DR

A progression in maths is a sequence of numbers in which each term follows a definite rule based on the previous term. The three classical types are arithmetic progression (AP, common difference ddd), geometric progression (GP, common ratio rrr), and harmonic progression (HP, reciprocals of an AP).

The 7-Year-Old Who Added 1 to 100 In 30 Seconds

In 1786, a German schoolteacher named Büttner asked his class to add the integers from 1 to 100. He expected to have a quiet hour. Then his 9-year-old student Carl Friedrich Gauss handed in the answer — 5050 — in roughly 30 seconds. Gauss had spotted the symmetry: pair the first with the last (1 + 100 = 101), the second with the second-last (2 + 99 = 101), and so on. There are 50 pairs, each summing to 101. Total: 50×101=5050. The trick generalises to the formula Sn=n2(a1+an) — the sum of any arithmetic progression. Every formula in this article is, at heart, a pairing or a multiplication trick that turns a long addition into a short one.

A progression in maths is a sequence in which every term after the first is obtained from the previous term by a fixed rule. The three classical progressions — arithmetic, geometric, and harmonic — are the cases where that rule is add a constant, multiply by a constant, or take a reciprocal-of-arithmetic.

Arithmetic Progression (AP) — The Constant Difference

An arithmetic progression is a sequence where each term differs from the previous one by a fixed value ddd, called the common difference.

General form: a1,a1+d,a1+2d,a1+3d,…

nnn-th term formula:

an=a1+(n−1)d

Sum of first nnn terms:

Sn=n2(2a1+(n−1)d)=n2(a1+an)

The two sum formulas are equivalent — use whichever fits the given information. If you know a1, d, and n, use the first. If you know a1 and an, use the second (it's literally Gauss's pairing trick generalised).

Examples: 5,9,13,17,… (a1=5, d=4). 20,15,10,5,0,−5,… (a1=20, d=−5).

Geometric Progression (GP) — The Constant Ratio

A geometric progression is a sequence where each term equals the previous term multiplied by a fixed value rrr, called the common ratio.

General form: a1,a1r,a1r2,a1r3,…

nnn-th term formula:

an=a1⋅rn−1

Sum of first nnn terms (for r≠1):

Sn=a1⋅rn−1r−1=a1⋅1−rn1−r

Sum of an infinite GP (only when ∣r∣<1):

S∞=a11−r

Examples: 3,6,12,24,… (a1=3, r=2). 81,27,9,3,1,13,… (a1=81, r=13). The second one has an infinite sum: S∞=812/3=121.5.

Harmonic Progression (HP) — The Reciprocal Of An AP

A harmonic progression is a sequence whose reciprocals form an arithmetic progression. There is no direct closed-form sum for an HP — to work with one, convert to its AP of reciprocals first.

General form: 1/a1, 1/(a1+d), 1/(a1+2d), 1/(a1+3d),…

nnn-th term formula: Take the reciprocal of the nnn-th term of the corresponding AP:

anHP=1/(a1AP+(n−1)d)

Example: 1,1/2,1/3,1/4,1/5,… — its reciprocals are the AP 1,2,3,4,5,…. The 100th HP term is 1/100.

The HP shows up in music (string-length ratios for harmonic intervals), in optics (lens equations), and in machine learning (the learning-rate decay schedule 1/t).

Quick — Standard — Stretch: three worked examples

Quick — find the 20th term of the AP 5,9,13,17,…

a1=5, d=4, n=20.

a20=5+(20−1)(4)=81

Final answer: a20=81

Standard (Wrong-Path-First) — find the 10th term of the harmonic progression 1,1/4,1/7,1/10,…

Correct method. Step 1: take the reciprocal of every term — get the AP 1,4,7,10,… . Step 2: find the 10th term of this AP: a10AP=1+(10−1)(3)=28. Step 3: take the reciprocal back: a10HP=1/28.

Final answer: a10=1/28

Stretch — find the sum of the infinite GP 1,2/3,4/9,8/27,…

a1=1, r=2/3.

S∞=1/(1−2/3)=3

Final answer: S∞=3.

Which Progression Do I Have? A 30-Second Diagnostic

Given a sequence, identify the type by checking three things in order:

Compute consecutive differences. Constant? It's an AP.
Compute consecutive ratios. Constant? It's a GP.
Take the reciprocals and re-test (1) and (2). Reciprocals form an AP? It's an HP.

Why Progressions Matter — From Finance To Music

Progressions sit underneath several practical models.

Compound interest. A balance growing at rate r each year is a geometric progression: P,P(1+r),P(1+r)^2,… . After n years, A=P(1+r)ⁿ.
Loan amortisation. Each monthly payment splits between principal and interest in proportions that change geometrically.
Music intervals. The fundamental frequency f and its harmonics 2f,3f,4f,… form an AP — but the string lengths that produce them form an HP.
Population doubling. Bacterial growth in a Petri dish: each generation doubles.
Half-life and decay. Radioactive decay is a GP with 0<r<1.

Where Students Lose Marks On Progressions

Mistake 1: Trying to find a common difference in a geometric progression (or a common ratio in an arithmetic progression)

Correct way: Run both checks in order.

Mistake 2: Using the nnn-th term formula but counting positions wrong

Correct way: a10=a1+(10−1)d.

Mistake 3: Applying the infinite-GP sum formula when ∣r∣≥1

Correct way: Check the convergence condition first.

Aryabhata and Gauss — A Short History

Aryabhata (476–550, India). In the Aryabhatiya (499 CE), Aryabhata gave compact rules for the sum of an arithmetic progression.

Carl Friedrich Gauss (1777–1855, Germany). The schoolboy pairing trick produced the formula Sn=n2(a1+an) that every AP sum problem still uses.

Why it matters. Aryabhata gave the sum rules a precise algorithmic form; Gauss gave them the symbolic compactness that turned them into the formulas in every textbook.

Bottom Line

A progression in maths is a sequence whose terms follow a fixed rule from the previous term.
The three classical types are arithmetic (common difference), geometric (common ratio), and harmonic (reciprocals form an AP).
AP n-th term: an=a1+(n−1)d. GP n-th term: an=a1rn−1. HP: solve in the AP of reciprocals.
The infinite GP sum S∞=a1/(1−r) works only when ∣r∣<1 — otherwise the sum diverges.

Sharpen your progression skills — three practice problems

Find the sum of the first 50 terms of the AP 4,9,14,19,…
The third term of a GP is 12 and the sixth is 96. Find the first term, the common ratio, and the sum to infinity (if it exists).
The first three terms of an HP are 12,1/6,1/10. Find the 12th term.