Arithmetic Progression

An arithmetic progression (AP) is a sequence in which each term differs from the previous one by a fixed amount called the common difference. Unlike a geometric progression where you multiply by a constant ratio, an AP grows by repeated addition. APs describe situations with linear, steady change: equal monthly savings, stair steps of fixed height, the days in successive months of a year, simple-interest growth, and rows of seats that gain a fixed number from front to back.

Definition and General Term

An arithmetic progression has the form: $ a, a+d, a+2d, a+3d, \ldots $, where $ a $ is the first term and $ d $ is the common difference. The $ n $-th term is:

$$ a_n = a + (n – 1)\, d $$

A sequence is an AP precisely when the difference between consecutive terms is constant: $ a_2 – a_1 = a_3 – a_2 = \cdots = d $.

Worked Examples

3, 7, 11, 15, 19, $ \ldots $: $ a = 3 $, $ d = 4 $. 20th term: $ 3 + 19 \cdot 4 = 79 $.
100, 95, 90, 85, $ \ldots $: $ a = 100 $, $ d = -5 $. Decreasing AP. 21st term: $ 100 + 20 \cdot (-5) = 0 $.
$ \tfrac{1}{2}, 1, \tfrac{3}{2}, 2, \ldots $: $ a = \tfrac{1}{2} $, $ d = \tfrac{1}{2} $. 15th term: $ \tfrac{1}{2} + 14 \cdot \tfrac{1}{2} = \tfrac{15}{2} = 7.5 $.

Sum of an AP

The sum of the first $ n $ terms of an AP is:

$$ S_n = \frac{n}{2}\,[2a + (n-1)d] = \frac{n}{2}\,(a + a_n) $$

The second form has an elegant geometric interpretation: the sum equals the number of terms times the average of the first and last terms.

The Story of Gauss

Carl Friedrich Gauss, age 7 or 8, was reportedly told by his teacher to sum the integers from 1 to 100 — a task meant to keep the class occupied. Gauss noticed that pairing 1 + 100, 2 + 99, 3 + 98, all equal 101, and there are 50 such pairs, so the total is 50 · 101 = 5050. He had the answer almost instantly. The story is the most famous illustration of the AP sum formula.

Arithmetic Mean

In any AP, each term (except the first and last) is the arithmetic mean of its neighbors: $ a_n = \tfrac{1}{2}(a_{n-1} + a_{n+1}) $. The arithmetic mean of two numbers $ x $ and $ y $ is $ (x+y)/2 $ — the most familiar ‘average’ in everyday usage.

AP vs GP: Side-by-Side

Property	Arithmetic progression	Geometric progression
Operation between terms	Addition (constant d)	Multiplication (constant r)
nth term	a + (n-1)d	a · r^(n-1)
Sum of n terms	n/2 · [2a + (n-1)d]	a · (1 – rⁿ)/(1 – r)
Type of growth	Linear	Exponential
Sequence example	3, 7, 11, 15, …	3, 12, 48, 192, …

Applications

Simple interest. Money invested at simple interest grows arithmetically: P, P + Pr, P + 2Pr, P + 3Pr, … — an AP with common difference Pr.
Salary increments. A fixed annual raise of a set dollar amount produces an AP of yearly earnings.
Stadium seating. Stadium rows often gain a fixed number of seats per row; the row counts form an AP.
Distance with constant deceleration. An object decelerating uniformly covers distances in successive seconds that form an AP. (Galileo proved this geometrically.)
Counting and combinatorics. The sum 1+2+…+n is the count of handshakes between n+1 people, the number of edges in a complete graph, and the closed form n(n+1)/2.

Frequently Asked Questions

What is an arithmetic progression?

A sequence in which each term is obtained by adding a fixed number (the common difference) to the previous term. Example: 5, 9, 13, 17, … has common difference 4. The general formula for the nth term is a_n = a + (n−1)d.

What’s the difference between an arithmetic and geometric progression?

An arithmetic progression has a constant difference between consecutive terms (you add d each step). A geometric progression has a constant ratio (you multiply by r each step). 3, 7, 11, 15 is arithmetic with d = 4. 3, 6, 12, 24 is geometric with r = 2. APs grow linearly, GPs grow exponentially.

How do you find the sum of an AP?

S_n = (n/2) × [2a + (n−1)d], or equivalently S_n = (n/2) × (a + a_n) where a_n is the last term. The second form is easier to remember: the sum equals the count of terms times the average of the first and last terms.

How did Gauss sum 1 to 100?

By pairing 1 + 100, 2 + 99, 3 + 98, and so on. Each pair sums to 101, and there are 50 such pairs. So the total is 50 × 101 = 5050. He did this as a child to finish a tedious classroom exercise quickly. The trick generalises into the AP sum formula S_n = (n/2)(a + a_n).

Can an arithmetic progression be decreasing?

Yes — if the common difference d is negative. Example: 100, 95, 90, 85, … has d = −5. Decreasing APs include cooling temperatures by a constant amount per minute, scoring penalties that subtract a fixed amount per missed shot, or stockpile depletion at a fixed rate.

What is the arithmetic mean?

The arithmetic mean of two or more numbers is their sum divided by how many there are. The most familiar ‘average’. In an AP, every middle term equals the arithmetic mean of its immediate neighbors: a_n = (a_(n−1) + a_(n+1))/2. This is where the AP gets its name.