Foundations — Geometric progression (GP) — nth term, sum of n terms — derivations

This page assumes you know nothing beyond counting and basic arithmetic. We will earn every letter and every little raised number before we let the parent note use it. Read top to bottom — each block is a rung, and the next rung needs the one below it.

0. Two ways a sequence can grow — the picture first

A sequence is just numbers written in order, one after another, like beads on a string. Nothing fancy — $2,5,8,11$ is a sequence; so is $2,4,8,16$.

There are two famous ways to make the next bead from the current one:

• Add the same amount each step → this is an Arithmetic Progression (AP) (a ladder with equal rungs).
• Multiply by the same amount each step → this is our GP (a photocopier that resizes).

1. The symbol $a$ — the first term

The parent sometimes writes it $a_1$ (read "a-sub-one"). The little $1$ below is called a subscript — it is not multiplication and not a power. It is a house number telling you which term we mean.

2. The symbol $r$ — the common ratio

How do you find $r$ from a given sequence? You divide a term by the one before it. That division is the heart of the word "ratio."

3. Powers and exponents — the shorthand for "multiply $r$ by itself"

Since a GP multiplies by $r$ again and again, we need a compact way to write "$r$ times $r$ times $r$ …". That shorthand is a power.

The single most important special case — and the one that makes the nth-term formula click:

Putting the last three ideas together — start at $a$, multiply by $r$ once per arrow, and count $n-1$ arrows — gives the parent note's headline result, the nth term:

Two power facts the parent quietly uses when solving for $n$:

When the two powers can't be matched by eye, you reach for Logarithms — the tool that asks "what exponent produces this number?" That is the topic's escape hatch for solving $r^{n-1}=k$.

4. Sigma-free summation — what "$S_n$" and the "$\dots$" really mean

The three dots "$\cdots$" (an ellipsis) mean "keep going with the obvious pattern, I'm not writing every term." In $a+ar+a{r}^2+\cdots +a{r}^{n-1}$ the dots stand in for all the middle terms that follow the same $\times r$ rule.

5. Algebra you'll lean on: factoring and division-by-zero

The parent's sum derivation uses two moves. Make sure both are solid.

6. Where these feed the topic — prerequisite map

Every box on the left is something this page built from zero; every arrow shows it flowing into the parent note's two headline results — the nth term $a_n=a{r}^{n-1}$ and the sum of $n$ terms $S_n=\frac{a(1-r^n)}{1-r}$.

7. All cases in one glance (so no scenario surprises you)

• $r>1$ (e.g. $r=2$): terms grow; jumps get bigger. Sum form $\frac{a(r^n-1)}{r-1}$ keeps both parts positive.
• $r=1$: every term equals $a$; the sequence is flat. Sum is $na$ (the general formula would divide by zero).
• $0<r<1$ (e.g. $r=\frac{1}{2}$): terms shrink toward $0$. Sum form $\frac{a(1-r^n)}{1-r}$ keeps both parts positive. As you take more and more terms this heads toward a fixed ceiling — the Sum of infinite GP.
• $r=-1$ (e.g. $a,-a,a,-a,\dots$): a perfectly alternating sequence that never grows or shrinks — it just flips between two values. Note $(-1{)}^p=(-1{)}^q$ only forces $p$ and $q$ to match in evenness (same remainder when divided by 2), not $p=q$, so the "equal powers ⇒ equal exponents" rule from §3 does not apply here. The sums also just flip: $S_n$ is $a$ when $n$ is odd and $0$ when $n$ is even.
• $r<0$, $r\ne -1$ (e.g. $r=-2$): terms flip sign every step and change size — the string zig-zags above and below zero with growing swings. The formulas still hold; just carry the minus signs carefully.
• Forbidden: $r=0$ (every term after the first is $0$) and $a=0$ (the whole sequence is $0$). Both are ruled out so that "divide by a term" always makes sense.

Equipment checklist

Self-test: can you say each without peeking?

Connections

• Parent note (Hinglish) — the derivations these foundations unlock.
• Arithmetic Progression (AP) — the "add" cousin; contrast subtract-for-$d$ vs divide-for-$r$.
• Exponential functions — powers $r^k$ made continuous.
• Logarithms — the tool that inverts a power to solve for $n$.
• Sum of infinite GP — where $0<r<1$ leads when $n$ keeps growing.
• Geometric Mean — the middle term of a 3-term GP.
• Compound interest — money whose value is a GP with $r=1+\frac{i}{100}$.