Geometric Progressions

⚡ Key Takeaways

A geometric progression (GP) is a sequence in which each term is obtained by multiplying the previous one by a fixed nonzero number called the common ratio $r$ . Key result: the $n$ -th term is $a_n = a\,r^{n-1}$ , and the sum of $n$ terms is $S_n = a\frac{r^n-1}{r-1}$ . GPs model compound interest, population growth, radioactive decay, and any process with constant multiplicative change. Studied since antiquity (Euclid, ~300 BCE); foundational to exponential functions and series.

💡 The Core Insight

Repeated multiplication, not repeated addition. In an arithmetic progression you walk forward by equal steps; in a geometric progression you scale by equal factors. Because scaling compounds, GP terms grow or shrink exponentially — and that single multiplicative rule generates the entire structure: every term, every sum, and the convergence behaviour of the infinite series all follow from the constant ratio $r$ .

DefinitionGeometric Progression

A geometric progression is a sequence $(a_n)_{n\ge 1}$ such that there exists a fixed nonzero constant $r$ (the common ratio) with

1

\quad \frac{a_{n+1}}{a_n} = r \quad\text{for all } n\ge 1.

Derivation hint: This is the defining property — divide any term by its predecessor and the answer is always the same constant. Here:

$a = a_1$ — the first term (any nonzero real, sometimes complex),
$r$ — the common ratio ( $r \neq 0$ ; if $r=0$ all later terms vanish),
$n$ — the position index (a positive integer),
$a_n$ — the $n$ -th term.

Unrolling the recurrence gives the general term:

2

\quad a_n = a\,r^{\,n-1}.

Derivation hint: Multiply by $r$ exactly $n-1$ times starting from $a_1=a$ .

TheoremSum of a Finite Geometric Progression

For $r \neq 1$ , the sum of the first $n$ terms is

3

\quad S_n = \sum_{k=1}^{n} a\,r^{\,k-1} = a\,\frac{r^n - 1}{r - 1} = a\,\frac{1 - r^n}{1 - r}.

Derivation hint: Multiply $S_n$ by $r$ , subtract, and watch all interior terms cancel. When $r = 1$ every term equals $a$ , so $S_n = na$ .

TheoremSum of an Infinite Geometric Series

If $|r| < 1$ , the partial sums converge and

4

\quad S_\infty = \sum_{k=1}^{\infty} a\,r^{\,k-1} = \frac{a}{1 - r}.

Derivation hint: In equation [3] let $n\to\infty$ ; since $|r|<1$ , the term $r^n \to 0$ . If $|r| \ge 1$ (and $a \neq 0$ ) the series diverges — it has no finite sum.

ProofDeriving the Finite Sum Formula

We prove equation [3] by the classic shift-and-subtract method.

Step 1. Write the sum explicitly:

5

\quad S_n = a + ar + ar^2 + \cdots + ar^{\,n-1}.

Derivation hint: Just list all $n$ terms from equation [2].

Step 2. Multiply both sides by the common ratio $r$ :

6

\quad r\,S_n = ar + ar^2 + ar^3 + \cdots + ar^{\,n}.

Derivation hint: Distribute $r$ across each term; every exponent rises by one.

Step 3. Subtract [6] from [5]. All the interior terms ( $ar$ through $ar^{n-1}$ ) appear in both lines and cancel:

7

\quad S_n - r\,S_n = a - a\,r^{\,n}.

Derivation hint: Only the very first term of [5] and the very last term of [6] survive.

Step 4. Factor each side:

8

\quad S_n(1 - r) = a(1 - r^{\,n}).

Derivation hint: Pull out $S_n$ on the left, $a$ on the right.

Step 5. Divide by $(1-r)$ , valid because $r \neq 1$ :

9

\quad S_n = a\,\frac{1 - r^{\,n}}{1 - r}.

Derivation hint: Division is legal precisely when $r\neq1$ ; the $r=1$ case is handled separately as $S_n=na$ . Multiplying top and bottom by $-1$ gives the equivalent form $a\frac{r^n-1}{r-1}$ . $\blacksquare$

∎

ProofConvergence of the Infinite Series for |r|<1

Starting from [9], examine the behaviour of $r^n$ as $n$ grows.

Step 1. If $|r| < 1$ , then $|r^n| = |r|^n \to 0$ as $n \to \infty$ .

10

\quad \lim_{n\to\infty} r^{\,n} = 0 \quad (|r| < 1).

Derivation hint: Repeated multiplication by a number of magnitude below 1 shrinks toward zero.

Step 2. Substitute this limit into [9]:

11

\quad S_\infty = \lim_{n\to\infty} a\,\frac{1 - r^{\,n}}{1 - r} = a\,\frac{1 - 0}{1 - r} = \frac{a}{1 - r}.

Derivation hint: The limit of a sum/quotient is the sum/quotient of the limits when the denominator is fixed and nonzero. If $|r|\ge 1$ , $r^n$ does not approach $0$ , so the partial sums do not settle — the series diverges. $\blacksquare$

∎

mathematician · Euclid of Alexandria | c. 325–265 BCE

Euclid gave one of the earliest rigorous treatments of geometric progressions in Book IX of the Elements</strong>. Proposition IX.35 states the finite-sum result in geometric language equivalent to equation [3], and IX.36 uses GPs to characterise perfect numbers (numbers equal to the sum of their proper divisors) via the formula $2^{p-1}(2^p-1)$ when $2^p-1$ is prime. His proof of the sum used proportion theory rather than algebra, but the shift-and-subtract idea is already present in spirit.

🧠 Intuition

Picture standing at position $a$ and being told: "each second, scale your value by $r$ ." If $r=2$ you double — explosive growth ( $a, 2a, 4a, 8a,\dots$ ). If $r=\tfrac12$ you halve — decay toward zero ( $a, \tfrac a2, \tfrac a4,\dots$ ). If $r=1$ you stand still. If $r$ is negative the sign flips each step, so the terms oscillate while their magnitudes follow $|r|^{n-1}$ . For the infinite sum with $|r|<1$ , think of repeatedly adding half the remaining gap to a target. Adding $\tfrac12 + \tfrac14 + \tfrac18 + \cdots$ keeps closing the distance to $1$ but never overshoots, because each piece is exactly half of what's left. The total is finite ( $=1$ ) even though there are infinitely many pieces — the pieces shrink fast enough. That "fast enough" is precisely the condition $|r|<1$ .

Figure

Geometric progression with

a=1

,

r=\tfrac12

: terms shrink, partial sums climb toward

S_\infty=2

.

ExampleFinite Sum — A Compound-Interest-Style Doubling

Given: A culture starts with $a = 500$ bacteria and the population multiplies by $r = 3$ every hour.

Question 1 — How many bacteria after 6 hours (i.e. the 7th term, counting the start as $a_1$ )? Use equation [2] with $n = 7$ :

a_7 = 500 \cdot 3^{\,7-1} = 500 \cdot 3^{6} = 500 \cdot 729 = 364{,}500.

Question 2 — Total bacteria-hours summed over the first 7 readings (hours 0 through 6)? Use equation [3] with $a=500$ , $r=3$ , $n=7$ :

S_7 = 500 \cdot \frac{3^{7} - 1}{3 - 1} = 500 \cdot \frac{2187 - 1}{2} = 500 \cdot 1093 = 546{,}500.

Answer:

\boxed{a_7 = 364{,}500 \text{ bacteria}, \qquad S_7 = 546{,}500 \text{ bacteria-hours}}

ExampleInfinite Sum — A Bouncing Ball's Total Distance

Given: A ball is dropped from height $h_0 = 10\text{ m}$ . After each bounce it rebounds to a fraction $r = 0.6$ of its previous height. Find the total vertical distance travelled before it comes to rest.

Step 1 — The first drop: $10\text{ m}$ down (no rebound before it).

Step 2 — Each subsequent bounce contributes both an up and a down trip. The rebound heights form a GP: $10(0.6), 10(0.6)^2, \dots$ Each rebound height is travelled twice (up then down). The total of all up-and-down rebound distances is:

D_{\text{rebound}} = 2\sum_{k=1}^{\infty} 10\,(0.6)^k = 2 \cdot \frac{10(0.6)}{1 - 0.6} = 2 \cdot \frac{6}{0.4} = 2 \cdot 15 = 30\text{ m}.

(Here the GP starts at first term $10(0.6)=6$ with ratio $0.6$ , so $S_\infty = 6/0.4 = 15$ , by equation [4].)

Step 3 — Add the initial drop:

D_{\text{total}} = 10 + 30 = 40\text{ m}.

Answer:

\boxed{D_{\text{total}} = 40 \text{ m}}

⚠️ Common mistake

The formula $S_n = a\frac{r^n-1}{r-1}$ fails when $r=1$ — you would divide by zero. When $r=1$ every term is $a$ , so the sum is simply $S_n = na$ . Always check the value of $r$ before applying the formula.

The infinite-sum formula $\frac{a}{1-r}$ requires $|r|<1$ . Plugging in $r=2$ to get $\frac{a}{1-2}=-a$ is a meaningless number — the actual series $a+2a+4a+\cdots$ diverges to infinity. The formula's algebra is happy to produce an answer that is not the sum.

Counting terms vs. counting ratios. Going from $a_1$ to $a_n$ involves $n-1$ multiplications, hence the exponent $n-1$ in equation [2], not $n$ . Off-by-one errors here are the single most common GP mistake.

CounterexampleA Divergent Series the Formula "Computes" Anyway

Consider $1 + 2 + 4 + 8 + 16 + \cdots$ with $a=1$ , $r=2$ . Blindly applying equation [4]:

\frac{a}{1-r} = \frac{1}{1-2} = -1.

This is famously the value assigned by certain analytic continuation methods (2-adic / Abel-type summation), but in the ordinary sense the partial sums $1,3,7,15,31,\dots$ march off to $+\infty$ . The series does not converge, and $-1$ is not its sum in the standard real-number sense. The lesson: a formula derived under a hypothesis ( $|r|<1$ ) may still output a number when the hypothesis fails — that number is not the answer to the original question.

connection

Geometric progressions sit at the crossroads of several core ideas:

They are the discrete cousins of the Exponential Function: $a_n = a r^{n-1}$ is just $f(x)=a r^{x-1}$ sampled at integers.
Taking logarithms turns a GP into an Arithmetic Progression: $\log a_n = \log a + (n-1)\log r$ is linear in $n$ .
The infinite-sum result is a first encounter with Convergence of Series and the notion of a limit.
The geometric mean $\sqrt{a_{n-1}a_{n+1}} = a_n$ links GPs to the Means and Inequalities (AM–GM).
They underpin Compound Interest and the Binomial Theorem's expansion of $(1+x)^n$ relates to truncated geometric-type sums.

🔬 Where this appears

⏳ History

- c. 1650 BCE — Rhind Mathematical Papyrus (Egypt): Problem 79 sums a geometric series of houses, cats, mice, etc. — an early "geometric" word problem.

- c. 300 BCE — Euclid, Elements Book IX: Propositions IX.35–36 prove the finite-sum result and connect GPs to perfect numbers.

- c. 250 BCE — Archimedes: In Quadrature of the Parabola, he sums

1+\tfrac14+\tfrac1{16}+\cdots=\tfrac43

to find the area under a parabola — an infinite geometric series used rigorously.

- 5th c. BCE — Zeno of Elea: His dichotomy paradox implicitly poses the convergence of

\tfrac12+\tfrac14+\cdots

, resolved millennia later by limit theory.

- 14th c. CE — Nicole Oresme & the Merton School: Studied convergent and divergent series; Oresme proved the harmonic series diverges, sharpening the contrast with convergent GPs.

- 1593 — François Viète: Published an explicit formula for the sum of an infinite geometric series.

- 17th c. — Newton & Leibniz: Geometric series became a workhorse of the new calculus (e.g.

\frac{1}{1-x}=\sum x^n

as a power series).

- 18th c. — Euler: Manipulated geometric and related series freely, including divergent ones, anticipating modern summation methods.

- 19th c. — Cauchy & Weierstrass: Placed series convergence on rigorous footing, formalising exactly when

\frac{a}{1-r}

is legitimate (

|r|<1

).

open-problem

The clean dichotomy " $|r|<1$ converges, otherwise diverges" hides a rich frontier. Divergent series like $1+2+4+\cdots$ can be assigned consistent finite "sums" by methods such as Abel, Cesàro, $p$ -adic, or zeta-function regularisation — tools that are not folklore tricks but appear in quantum field theory (the notorious $1+2+3+\cdots = -\tfrac1{12}$ ). Determining which summation method is physically or mathematically appropriate, and when different methods agree, remains a deep theme in analysis. Likewise, the boundary case $|r|=1$ (e.g. $r=-1$ : $1-1+1-1+\cdots$ , Grandi's series) sits exactly on the edge of convergence and forces careful definitions of what "sum" even means. Extending GP ideas to complex ratios $r\in\mathbb C$ opens the door to power series, radii of convergence, and the analytic functions of complex analysis.

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also