Q.If the AM of two unequal positive real numbers $a$ and $b$ ($a > b$) is twice as much as their G.M., show that $a : b = (2+\sqrt{3}) : (2-\sqrt{3})$.

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 …