Q.One side of an equilateral triangle is 24 cm. The mid points of its sides are joined to form another triangle whose mid points are joined to form yet another triangle, and so on. This process continues indefinitely. Find the sum of the perimeters of all the triangles.

Sequences: From Intuition to Definition

Think of a sequence as a numbered list that goes on forever. You already know lists — a grocery list, a to-do list. A sequence is just that, but with two special rules: every item gets a natural number label (1st, 2nd, 3rd, …), and the list never ends.

For example, the odd numbers: 1, 3, 5, 7, 9, … That’s a sequence. The 1st term is 1, the 2nd is 3, the 3rd is 5, and so on. The "…" means it keeps going — there is no last term.

The key intuition: a sequence is an ordered, infinite parade of numbers, where each number has a fixed position.

The Precise Definition

A sequence is a function whose domain is the set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$ (or sometimes $\{0, 1, 2, \dots\}$) and whose range is a set of real numbers.

We write the sequence as $\{a_n\}_{n=1}^\infty$ or simply $(a_n)$, where $a_n$ is the $n$th term of the sequence.

How We Describe a Sequence

There are three common ways to specify a sequence:

1. Listing the first few terms — enough to show the pattern.

Example: $2, 4, 6, 8, 10, \dots$ (the even numbers).

2. A formula for the $n$th term — the most powerful method.

Example: $a_n = 2n$. Then $a_1 = 2$, $a_2 = 4$, $a_3 = 6$, and so on.

3. A recurrence relation — each term is defined using previous terms, plus a starting value.

Example: $a_1 = 1$, and $a_n = a_{n-1} + 2$ for $n \ge 2$. This also gives $1, 3, 5, 7, \dots$.

Why This Matters …