Arithmetic Progression (AP) and Arithmetic Mean

A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$ written in a definite order and following some rule. Each number is a term; $a_n$ is the general term (or $n$th term) and the rule that produces it is the defining formula. A sequence is finite if it stops after a fixed number of terms and infinite if it goes on forever. When you add the terms of a sequence you get a series: from the sequence $a_1, a_2, \dots, a_n$ comes the series $a_1 + a_2 + \dots + a_n$, whose value is written $S_n$.

Reading a sequence from its $n$th-term rule. If you are told $a_n = 2n + 3$, you generate terms by substituting $n = 1, 2, 3, \dots$ — giving $5, 7, 9, \dots$. This is the single most basic skill in the chapter: a compact formula stands in for the whole infinite list, exactly the way set-builder notation stands in for a set.

An Arithmetic Progression (AP) is the simplest pattern of all: each term is obtained by adding a fixed number to the one before it. That fixed number is the common difference $d$, and the starting value is the first term $a$. So an AP looks like $a,\ a+d,\ a+2d,\ a+3d, \dots$. The defining test is simple — if $a_{n} - a_{n-1}$ is the same constant for every $n$, the sequence is an AP. The general ($n$th) term follows directly:

To add up the first $n$ terms you use the sum formula. There are two equivalent versions — use whichever fits the information you are given:

Here $\ell = a_n$ is the last term. The second form is Gauss's pairing idea — pair the first and last terms, the second and second-last, and so on; each pair sums to $a + \ell$, and there are $\tfrac{n}{2}$ such pairs. A handy bridge between the two formulas is that the term and the sum are linked by $a_n = S_n - S_{n-1}$, which lets you recover any term from the sums.

The Arithmetic Mean (AM) of two numbers $a$ and $b$ is the number $A$ that sits exactly between them so that $a, A, b$ form an AP. Equating the common differences ($A - a = b - A$) gives:

More generally, to insert $k$ arithmetic means between $a$ and $b$ means to find $k$ numbers that, placed between them, make the whole list an AP of $k+2$ terms. Since $b$ is then the $(k+2)$th term, the common difference is $d = \dfrac{b-a}{k+1}$, and the $i$th inserted mean is $A_i = a + i\,d$. A neat consequence: the sum of all $k$ inserted means equals $k$ times the single AM of $a$ and $b$, i.e. $k\cdot\dfrac{a+b}{2}$.

It helps to see the AP machinery beside its multiplicative cousin, the GP, which you meet in the next topic:

Deeper Insight — why the AP is the natural home of constant change: An AP is the discrete version of a straight line. Plot the terms against their position $n$ and you get equally spaced points lying on a line of slope $d$ — that is the whole geometric meaning of "common difference". This is why APs model anything growing by a fixed amount each step: simple interest, a salary that rises by a flat increment, seats increasing row by row in an auditorium. The single most useful problem-solving trick in this topic is choosing terms symmetrically about the middle. When three numbers are in AP, write them as $a-d,\ a,\ a+d$ so the $d$'s cancel when you add; for four terms use $a-3d, a-d, a+d, a+3d$ with common difference $2d$. This symmetry turns ugly simultaneous equations into one-line solutions, and it is the reason examiners can pose "three numbers in AP" puzzles that look hard but collapse instantly once you set them up the right way.