Arithmetic Progression
Calculate the general term and sum of an Arithmetic Progression. Enter the first term, common difference and number of terms.
Last term (aₙ)
Sum of AP (Sₙ)
Sequence terms:
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.
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Calculate an arithmetic progression (AP)
In an arithmetic progression, each term grows out of the previous one plus a fixed value called the common difference. That's the case with 2, 5, 8, 11, and so on. From the data you enter, the tool calculates both the general term (any position in the sequence) and the sum of the terms.
Just indicate the first term, the common difference and how many terms you want; the AP formulas are applied next. It works for solving maths exercises, understanding sequences, adding up values that would grow too large on paper or checking a problem before a test.
The processing happens inside your browser. A direct reference for the AP: general term and sum solved, without you having to add value by value.