Sequences and Series

A sequence is an ordered list of numbers $a_1, a_2, a_3, \dots$ following some rule; a series is what you get when you add the terms of a sequence. 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 (nth) term follows directly from this:

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.

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 $d = \dfrac{b-a}{k+1}$.

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.

A Geometric Progression (GP) is built by multiplying, not adding. Each term is the previous one times a fixed number called the common ratio $r$. So a GP reads $a,\ ar,\ ar^2,\ ar^3, \dots$, where $a$ is the first term and $r \ne 0$. The test for a GP is that the ratio $\dfrac{a_n}{a_{n-1}}$ is the same for every $n$.

The general (nth) term records how many times you have multiplied by $r$:

The sum of the first $n$ terms has two forms; pick the one that keeps the numerator positive:

When $r = 1$ every term equals $a$, so $S_n = na$ — do not use the formula then, since it divides by zero.

Something remarkable happens when the ratio is a proper fraction. If $|r| < 1$, the terms shrink toward zero and the running total settles on a finite limit even though the series never stops. This is the sum to infinity:

The Geometric Mean (GM) of two positive numbers $a$ and $b$ is the value $G$ that makes $a, G, b$ a GP. Equating ratios ($\tfrac{G}{a} = \tfrac{b}{G}$) gives $G^2 = ab$, so:

There is a clean and famous relationship between the two means of the same pair of positive numbers — the AM–GM inequality:

Equality holds only when $a = b$. The proof is one line: $\left(\sqrt{a} - \sqrt{b}\right)^2 \ge 0$ expands to $a + b \ge 2\sqrt{ab}$.

Deeper Insight — multiplication, growth, and the surprise of a finite infinite sum: Where the AP is the discrete cousin of a straight line, the GP is the discrete cousin of exponential growth and decay. A constant ratio means each term is a fixed percentage larger (or smaller) than the last, which is exactly how compound interest, population growth, radioactive decay, and bouncing balls behave in the real world. The deepest idea in this topic is the sum to infinity. It feels paradoxical that adding infinitely many positive numbers can give a finite answer, but the resolution is that when $|r| < 1$ the terms vanish so fast that the leftover "tail" you keep adding becomes negligibly small — the partial sums creep ever closer to $\tfrac{a}{1-r}$ without ever overshooting it. This single mechanism is what lets a recurring decimal like $0.\overline{3}$ equal the exact fraction $\tfrac{1}{3}$. Finally, the AM–GM inequality is one of the most powerful tools in all of mathematics for proving things are as large or as small as possible; remembering that the two means coincide only when the numbers are equal is the key that unlocks countless optimisation arguments.

Beyond APs and GPs, three particular sums appear so often that their closed formulas are worth memorising. They are the sums of the first $n$ natural numbers, their squares, and their cubes. A special series is one whose terms follow a simple power pattern, letting us replace a long addition with a single tidy expression.

Sum of the first $n$ natural numbers — this is just an AP with $a = 1$, $d = 1$, so Gauss's pairing gives:

Sum of the first $n$ squares:

Sum of the first $n$ cubes — strikingly, this is exactly the square of the first formula:

These three are the toolkit; their real power shows when you combine them. Because summation is linear, you can split and scale any polynomial term and apply each formula separately:

Here is the full reference set side by side:

The way to use these on an unfamiliar series is the nth-term method: find a formula $a_k$ for the general term, then sum it. For instance, the series $1\cdot 2 + 2\cdot 3 + 3\cdot 4 + \dots$ has $a_k = k(k+1) = k^2 + k$, so its sum is $\sum k^2 + \sum k$, which you can evaluate immediately from the table.

Deeper Insight — find the term, then the formulas do the rest: The single most transferable skill in this topic is learning to read a series, spot the rule generating its general term $a_k$, and rewrite that term as a combination of powers of $k$. Once a sum is expressed as $\sum(\alpha k^2 + \beta k + \gamma)$, the three closed forms turn an apparently hard problem into routine arithmetic — no clever trick required, just bookkeeping. The cube identity $\sum k^3 = (\sum k)^2$ deserves a moment of wonder: it says the sum of the first $n$ cubes is a perfect square, and not just any square but the square of the triangular number $\tfrac{n(n+1)}{2}$. This is the kind of unexpected harmony that makes the subject worth studying, and it has a beautiful geometric proof involving nested square frames. Mastering the nth-term method here pays off far beyond this chapter: the same idea underlies the method of differences, telescoping sums, and ultimately the definite integral, where summing infinitely many tiny terms is exactly what you will be doing in Class 12.