Definition and the general term
Learning Objectives
- State the definition of an arithmetic progression and identify the common difference.
- Prove and apply the closed-form formula for the \(n\)-th term.
- Recognise the arithmetic-mean characterisation of an arithmetic progression.
- Relate the sign of the common difference to monotonicity.
Motivation
Many quantities change by a fixed amount over equal periods of time. The temperature drops by two degrees each hour during the night; a savings account grows by the same fifty euros every month; a runner adds five hundred metres to her daily distance each week. In each case, the values form a sequence with a particularly simple structure — the gap between any two consecutive terms is always the same.
Sequences of this kind are called arithmetic progressions. They are the simplest sequences with non-trivial behaviour, and they are the discrete analogue of linear functions: where a linear function changes at a constant rate over a continuous interval, an arithmetic progression changes by a constant step at each integer index.
Arithmetic progression
A sequence \((a_n)_{n \geq 1}\) of real numbers is called an arithmetic progression if there exists a real number \(d\) such that
\[a_{n+1} - a_n = d \quad \text{for every } n \geq 1.\]
The number \(d\) is called the common difference of the progression. It may be positive, negative, or zero.
An arithmetic progression is therefore determined by two pieces of data: its first term \(a_1\) and its common difference \(d\). Once these are fixed, every other term is forced.
First examples
\(1,\ 4,\ 7,\ 10,\ 13,\ \ldots\) is an arithmetic progression with \(a_1 = 1\) and common difference \(d = 3\).
\(10,\ 8,\ 6,\ 4,\ 2,\ 0,\ -2,\ \ldots\) is an arithmetic progression with \(a_1 = 10\) and \(d = -2\).
\(5,\ 5,\ 5,\ 5,\ \ldots\) is an arithmetic progression with \(a_1 = 5\) and \(d = 0\). The condition \(a_{n+1} - a_n = d\) is satisfied with \(d = 0\), so constant sequences qualify as arithmetic progressions in the formal sense.
\(1,\ 2,\ 4,\ 8,\ 16,\ \ldots\) is not an arithmetic progression: the difference \(a_2 - a_1 = 1\) is not equal to the difference \(a_3 - a_2 = 2\).
The general term
Let \((a_n)_{n \geq 1}\) be an arithmetic progression with first term \(a_1\) and common difference \(d\). Then, for every \(n \geq 1\),
\[a_n = a_1 + (n - 1)\, d.\]
We proceed by induction on \(n \geq 1\).
Base case. When \(n = 1\), the formula reads \(a_1 = a_1 + (1 - 1)\, d = a_1\), which holds.
Inductive step. Suppose \(a_n = a_1 + (n - 1)\, d\) for some \(n \geq 1\). By the defining condition of an arithmetic progression, \(a_{n+1} = a_n + d\). Substituting the inductive hypothesis,
\[a_{n+1} = a_1 + (n - 1)\, d + d = a_1 + n\, d = a_1 + \big( (n+1) - 1 \big)\, d.\]
This is the formula at index \(n + 1\), completing the induction.
By the Principle of Mathematical Induction, the formula \(a_n = a_1 + (n - 1)\, d\) holds for every \(n \geq 1\).
A note on indexing
This is an introductory course, and indexing arithmetic progressions from \(n = 1\) keeps the first term in the natural position \(a_1\). In computer science and more advanced mathematics, sequences are usually indexed from \(n = 0\); the same progression is then described by
\[a_n = a_0 + n\, d \quad \text{for } n \geq 0,\]
which has the same shape but one fewer subtraction. The underlying sequence is identical; only the label attached to each term shifts by one. As always, the first thing to check when reading any expression for the \(n\)-th term is where the indexing begins.
Monotonicity from the common difference
The sign of the common difference determines whether an arithmetic progression is monotone, and in which direction. If \(d > 0\), then \(a_{n+1} - a_n = d > 0\), so the progression is strictly increasing. If \(d < 0\), it is strictly decreasing. If \(d = 0\), every term equals \(a_1\), and the progression is constant — both non-decreasing and non-increasing.
Boundedness follows the same rule from the opposite direction. A non-constant arithmetic progression is unbounded: if \(d > 0\), the terms grow beyond every bound as \(n\) grows; if \(d < 0\), they fall below every bound. Only the constant progression \(d = 0\) is bounded.
The arithmetic-mean characterisation
A sequence \((a_n)_{n \geq 1}\) of real numbers is an arithmetic progression if and only if, for every \(n \geq 2\),
\[a_n = \frac{a_{n-1} + a_{n+1}}{2}.\]
In words: each term (after the first) is the arithmetic mean of its two neighbours. This property is, in fact, the reason for the name arithmetic progression.
(⇒) Suppose \((a_n)\) is an arithmetic progression with common difference \(d\). Then for every \(n \geq 2\),
\[a_{n+1} = a_n + d \quad \text{and} \quad a_{n-1} = a_n - d.\]
Adding these two equalities, \(a_{n-1} + a_{n+1} = 2a_n\), and dividing by \(2\) gives the desired identity.
(⇐) Suppose \(a_n = \dfrac{a_{n-1} + a_{n+1}}{2}\) for every \(n \geq 2\). Rearranging, \(a_{n+1} - a_n = a_n - a_{n-1}\) for every \(n \geq 2\). This says that consecutive differences are all equal: the difference between \(a_2\) and \(a_3\) is the same as between \(a_1\) and \(a_2\), the difference between \(a_3\) and \(a_4\) is the same as between \(a_2\) and \(a_3\), and so on. Setting \(d = a_2 - a_1\), we have \(a_{n+1} - a_n = d\) for every \(n \geq 1\), which is the defining condition of an arithmetic progression.
Using the general term
Suppose an arithmetic progression has \(a_1 = 7\) and \(d = 4\). The general term is
\[a_n = 7 + (n - 1) \cdot 4 = 4n + 3.\]
So \(a_{10} = 43\), and \(a_{100} = 403\). Conversely, suppose we are told only that \(a_3 = 11\) and \(a_8 = 26\), and asked to recover \(a_1\) and \(d\). Using the general term, \(a_8 - a_3 = 5d\), so
\[5d = 26 - 11 = 15 \quad \Longrightarrow \quad d = 3.\]
Then \(a_3 = a_1 + 2d = a_1 + 6 = 11\), so \(a_1 = 5\). The progression is \(5,\ 8,\ 11,\ 14,\ \ldots\)
Connection to Computer Science
An arithmetic progression is the discrete counterpart of evenly spaced values. A loop that increments a counter by a fixed amount at each step traces out an arithmetic progression; a range of array indices stepping by a constant stride is one too. When we ask how long such a loop runs to reach a target value, we are asking for the smallest \(n\) with \(a_n \geq T\), which the general-term formula answers in a single calculation rather than by simulating each step.
The same idea underlies linear address arithmetic in memory: the \(n\)-th element of an array at base address \(a_1\) with element size \(d\) is located at \(a_1 + (n-1)\,d\) — the general term formula, applied to physical addresses. Wherever a quantity changes by a fixed amount per step, an arithmetic progression is in the background.
Is every linear formula an arithmetic progression?
Every arithmetic progression has a linear general term: \(a_n = a_1 + (n-1)d\) is of the form \(An + B\) with \(A = d\) and \(B = a_1 - d\). The converse also holds: if a sequence is given by \(a_n = An + B\) for constants \(A, B \in \mathbb{R}\), then \(a_{n+1} - a_n = A\) for every \(n\), so the sequence is an arithmetic progression with common difference \(d = A\) and first term \(a_1 = A + B\).
The two notions therefore coincide: arithmetic progressions are exactly the sequences with a linear general term in \(n\). This is a small theorem in itself, and it confirms the intuition that an arithmetic progression is the discrete analogue of a straight line. A line in continuous mathematics, sampled at the integers, gives an arithmetic progression; conversely, an arithmetic progression, plotted as points in the plane, lies on a straight line.
Exercises
Exercise 1
For each sequence, decide whether it is an arithmetic progression.
Exercise 2
An arithmetic progression has \(a_1 = 5\) and common difference \(d = 3\). Use the general term formula to fill in the requested values.
Exercise 3
An arithmetic progression has \(a_4 = 14\) and \(a_9 = 34\). Find its common difference and first term.
Exercise 4
Decide whether each statement is true or false.
Summary
An arithmetic progression is a sequence \((a_n)_{n \geq 1}\) with a fixed common difference \(d\), meaning \(a_{n+1} - a_n = d\) for every \(n\).
The progression is determined by its first term \(a_1\) and \(d\), and its general term is
\[a_n = a_1 + (n - 1)\, d.\]
The sign of \(d\) determines monotonicity: \(d > 0\) strictly increasing, \(d < 0\) strictly decreasing, \(d = 0\) constant. Every non-constant arithmetic progression is unbounded.
A sequence is an arithmetic progression if and only if each term (after the first) is the arithmetic mean of its two neighbours — the property that gives the family its name.
Arithmetic progressions are exactly the sequences with a linear general term: \(a_n = An + B\) for constants \(A, B \in \mathbb{R}\), with \(d = A\).