Definition, nth Term and Sum

An arithmetic progression (AP) is one of the most fundamental structures in mathematics: a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. This constant is called the common difference and is usually denoted d. For example, the sequence 3, 7, 11, 15, 19, … is an AP with first term a = 3 and common difference d = 4. Arithmetic progressions appear throughout mathematics, science, and everyday life — from the rungs of a ladder spaced equally apart to the pages of a book numbered consecutively. The study of APs dates back to ancient civilisations. The Rhind Mathematical Papyrus (c. 1650 BCE) contains problems involving arithmetic sequences, and the great mathematician Carl Friedrich Gauss famously summed the integers from 1 to 100 as a schoolboy by recognising the arithmetic structure and pairing terms cleverly. The two central results you need to master are the formula for the nth term, which lets you jump directly to any position in the sequence, and the formula for the sum of the first n terms, which lets you add up a block of the sequence without tedious individual addition. Understanding APs builds the foundation for geometric progressions, series convergence, and many counting and combinatorial arguments encountered later in mathematics. Once you can identify a common difference and apply the nth term and sum formulas fluently, you will be equipped to solve a wide range of problems from financial calculations and physics kinematics to competition mathematics.

Key takeaways

Recognising an Arithmetic Progression
The General (nth) Term Formula
Deriving the Sum of n Terms
Key Properties and Checks

Recognising an Arithmetic Progression

A sequence a₁, a₂, a₃, … is an AP if and only if the difference between consecutive terms is constant: aₙ₊₁ − aₙ = d for all n ≥ 1. The value d can be positive (increasing sequence), negative (decreasing sequence), or zero (constant sequence). To verify a sequence is an AP, compute each consecutive difference and check they are all equal. For example, 20, 15, 10, 5, 0, −5, … has d = −5 (each term decreases by 5), so it is an AP.

The General (nth) Term Formula

Starting from the first term a₁ and adding d repeatedly, the nth term is aₙ = a₁ + (n − 1)d. This is a linear function of n: plotting aₙ against n gives points on a straight line with slope d and vertical intercept a₁ − d. For instance, in the AP 5, 9, 13, 17, … we have a₁ = 5 and d = 4, so the nth term is aₙ = 5 + (n − 1)·4 = 4n + 1. The 20th term is a₂₀ = 4(20) + 1 = 81.

Deriving the Sum of n Terms

Let Sₙ = a₁ + a₂ + … + aₙ. Writing the sum forwards and backwards simultaneously: Sₙ = a₁ + (a₁+d) + (a₁+2d) + … + aₙ Sₙ = aₙ + (aₙ−d) + (aₙ−2d) + … + a₁ Adding each column: 2Sₙ = n(a₁ + aₙ), so Sₙ = n(a₁ + aₙ)/2. Since aₙ = a₁ + (n−1)d, we can substitute to get the fully expanded form: Sₙ = n/2 · [2a₁ + (n−1)d]. This is Gauss's pairing argument in algebraic form. Note that Sₙ is a quadratic function of n.

Key Properties and Checks

Several useful properties follow directly from the formulas. (1) The arithmetic mean of any two terms equidistant from the centre of a finite AP equals the middle term. (2) If three numbers a, b, c are in AP, then 2b = a + c (b is the arithmetic mean of a and c). (3) The sum of the first and last term equals the sum of the second and second-to-last term, and so on — this symmetry is exactly what Gauss exploited. (4) You can find a₁ and d from any two terms: if you know aₘ and aₙ, then d = (aₙ − aₘ)/(n − m).

The Key Equation

a_n = a_1 + (n-1)d \qquad S_n = \frac{n}{2}\bigl[2a_1 + (n-1)d\bigr]

a_n — nth term (same as sequence values): The value of the term at position n in the arithmetic progression.
a_1 — first term (same as sequence values): The starting value of the arithmetic progression.
n — term number (position) (dimensionless positive integer): The index giving the position of the term in the sequence; n ≥ 1.
d — common difference (same as sequence values): The fixed constant added to each term to obtain the next term; d = aₙ₊₁ − aₙ.
S_n — sum of first n terms (same as sequence values): The total obtained by adding the first n terms of the arithmetic progression.

Worked Example

Problem. An arithmetic progression has first term a₁ = 7 and common difference d = 3. (a) Find the 15th term. (b) Find the sum of the first 20 terms. (c) Which term of the AP equals parasitically 100?

a_{15} = 49, \quad S_{20} = 710, \quad n = 32

Answer. a₁₅ = 49; S₂₀ = 710; the 32nd term equals 100.

Figure

Diagram

Origins · 1 minCarl Friedrich Gauss and how this was discovered

Carl Friedrich Gauss

1777–1855 · German

As a child, Gauss is said to have stunned his teacher by instantly computing the sum 1 + 2 + 3 + … + 100 = 5050. He recognised that pairing the first and last terms (1+100), the second and second-to-last (2+99), and so on, each pair summing to 101, with 50 such pairs, gives 50 × 101 = 5050. This elegant pairing argument is precisely the derivation of the AP sum formula Sₙ = n(a₁ + aₙ)/2, and it illustrates the power of seeing mathematical structure rather than performing brute-force arithmetic. Gauss went on to become one of the greatest mathematicians in history, making profound contributions to number theory, statistics, astronomy, and physics — but the arithmetic series story remains one of the most celebrated anecdotes in all of mathematics.

Applications of APs — Directly extends this concept by applying the nth term and sum formulas to real-world problems such as financial savings, seating arrangements, and motion under uniform acceleration.
GP Terms and Sums — Geometric progressions are the natural counterpart to APs: instead of a common difference, there is a common ratio. Comparing the two structures deepens understanding of both.
Geometric Mean — Just as three terms in an AP satisfy 2b = a + c (arithmetic mean), three terms in a GP satisfy b² = ac (geometric mean). Contrasting the two means reinforces AP properties.
AM–GM Relationship — The arithmetic mean of an AP is a key idea here; the AM–GM inequality formalises how the arithmetic mean compares to the geometric mean, linking both progression types.
Definition of Sequences — APs are a special class of sequences. Reviewing the general definition of sequences, including recursive and explicit forms, provides the broader framework into which APs fit.
Binomial Expansion — The binomial expansion of (1+x)ⁿ involves summing n+1 terms, and recognising arithmetic patterns in exponents and coefficients helps in manipulating series effectively.
Counting Principle — Counting principles underlie many sum formulae; understanding how to count systematically complements the index-based reasoning used in AP sum derivations.