Arithmetic Progression

What is an Arithmetic Progression?

In an AP, you've got a sequence where the gap from one term to the next never changes. That fixed gap is the common difference r.

• General term: aₙ = a₁ + (n − 1) × r
• Sum: Sₙ = n × (a₁ + aₙ) ÷ 2

Arithmetic progression (AP)

An arithmetic progression is a sequence in which each term differs from the previous one by a constant common difference r. The general term is aₙ = a₁ + (n - 1)·r and the sum of the first n terms is Sₙ = (a₁ + aₙ)·n/2 = n·a₁ + n(n-1)·r/2. Example: for a₁ = 2, r = 3, n = 5, the terms are 2, 5, 8, 11, 14, with a₅ = 14 and S₅ = (2 + 14)·5/2 = 40. The median of a finite AP equals its central term (or the average of the two middle terms when n is even).

A famous anecdote credits the young Carl Friedrich Gauss (~1786) with summing 1 + 2 + ... + 100 in seconds: pair 1+100, 2+99, ..., 50+51 — fifty pairs each summing to 101, giving 50 · 101 = 5050. This pairing trick is exactly the proof of the closed-form sum above.

Applications

APs appear in finance — the SAC amortisation table (constant amortisation) generates an AP of installments with constant principal and decreasing interest; linear depreciation reduces an asset's value by a fixed amount each period; equal installments with fixed monetary correction also form an AP. They also count rows in arrangements (people in lines, seats in a stadium tier) and pace running or training programs that increase volume by a fixed amount each week.

FAQ

How do I find r if I know two terms? Use r = (aₘ - aₙ)/(m - n).

What is an increasing/decreasing/constant AP? If r > 0 the progression is increasing; r < 0 decreasing; r = 0 constant (all terms equal).

Difference from a geometric progression? In an AP terms differ by a constant added; in a geometric progression they differ by a constant multiplied (the ratio q).

Can n be a non-integer? No — n is the position (1, 2, 3, ...) and must be a positive integer.