Geometric Progressions

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.

A geometric progression is an ordered list of numbers $a, a r, a r^{2}, a r^{3}, \dots$ where the ratio between any two consecutive terms is constant. It contrasts with an arithmetic progression, where consecutive terms differ by a constant sum rather than a constant factor. GPs matter because multiplicative growth and decay are everywhere — money earning interest, bacteria dividing, signals attenuating — and the GP is the discrete skeleton of the exponential function. A surprising fact: an infinite geometric series can have a finite sum (e.g. $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots = 1$ ), which is precisely the resolution of Zeno's paradox of motion.

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

Formal Statement

Geometric Progression

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

\frac{a _{n + 1}}{a _{n}} = r for all n \geq 1.

⟳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:

a_{n} = a r^{n - 1} .

⟳Multiply by

r

exactly

n - 1

times starting from

a_{1} = a

Sum of a Finite Geometric Progression

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

S_{n} = k = 1 \sum n a r^{k - 1} = a \frac{r ^{n} - 1}{r - 1} = a \frac{1 - r ^{n}}{1 - r} .

⟳Multiply

S_{n}

r

, subtract, and watch all interior terms cancel.

When $r = 1$ every term equals $a$ , so $S_{n} = na$ .

Sum of an Infinite Geometric Series

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

S_{\infty} = k = 1 \sum \infty a r^{k - 1} = \frac{a}{1 - r} .

⟳In equation [3] let

n \to \infty

; since

∣ r ∣ < 1

, the term

r^{n} \to 0

If $∣ r ∣ \geq 1$ (and $a \neq = 0$ ) the series diverges — it has no finite sum.

Why It Holds / Derivation

Deriving the Finite Sum Formula

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

Step 1. Write the sum explicitly:

S_{n} = a + a r + a r^{2} + \dots + a r^{n - 1} .

⟳Just list all

n

terms from equation [2].

Step 2. Multiply both sides by the common ratio

r

r S_{n} = a r + a r^{2} + a r^{3} + \dots + a r^{n} .

⟳Distribute

r

across each term; every exponent rises by one.

Step 3. Subtract [6] from [5]. All the interior terms (

a r

through

a r^{n - 1}

) appear in both lines and cancel:

S_{n} - r S_{n} = a - a r^{n} .

⟳Only the very first term of [5] and the very last term of [6] survive.

Step 4. Factor each side:

S_{n} (1 - r) = a (1 - r^{n}) .

⟳Pull out

S_{n}

on the left,

a

on the right.

Step 5. Divide by

(1 - r)

, valid because

r \neq = 1

S_{n} = a \frac{1 - r ^{n}}{1 - r} .

⟳Division is legal precisely when

r \neq = 1

; the

r = 1

case is handled separately as

S_{n} = na

Multiplying top and bottom by $- 1$ gives the equivalent form $a \frac{r ^{n} - 1}{r - 1}$ . $■$

Convergence 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

n \to \infty

n \to \infty lim r^{n} = 0 (∣ r ∣ < 1) .

⟳Repeated multiplication by a number of magnitude below 1 shrinks toward zero.

Step 2. Substitute this limit into [9]:

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

⟳The limit of a sum/quotient is the sum/quotient of the limits when the denominator is fixed and nonzero.

If $∣ r ∣ \geq 1$ , $r^{n}$ does not approach $0$ , so the partial sums do not settle — the series diverges. $■$

Euclid of Alexandria | c. 325–265 BCE

Euclid gave one of the earliest rigorous treatments of geometric progressions in Book IX of the Elements. 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.

Building Intuition

Picture standing at position $a$ and being told: "each second, scale your value by $r$ ." If $r = 2$ you double — explosive growth ( $a, 2 a, 4 a, 8 a, \dots$ ). If $r = \frac{1}{2}$ you halve — decay toward zero ( $a, \frac{a}{2}, \frac{a}{4}, \dots$ ). 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 $\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$ 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 $∣ r ∣ < 1$ .

Geometric progression with $a=1$, $r=\tfrac12$: terms shrink, partial sums climb toward $S_\infty=2$.

Worked Examples

Finite 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:

a_{7} = 364, 500 bacteria, S_{7} = 546, 500 bacteria-hours

Infinite Sum — A Bouncing Ball's Total Distance

Given: A ball is dropped from height

h_{0} = 10 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 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_{rebound} = 2 k = 1 \sum \infty 10 (0.6)^{k} = 2 \cdot \frac{10 ( 0.6 )}{1 - 0.6} = 2 \cdot \frac{6}{0.4} = 2 \cdot 15 = 30 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_{total} = 10 + 30 = 40 m .

Answer:

D_{total} = 40 m

Common Misconceptions

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 + 2 a + 4 a + \dots

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}

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.

A Divergent Series the Formula "Computes" Anyway

Consider $1 + 2 + 4 + 8 + 16 + \dots$ 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.

Connections

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: $lo g a_{n} = lo g a + (n - 1) lo g r$ 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 $a_{n - 1} a_{n + 1} = a_{n}$ 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.

Real-World Applications

Compound interest & savings. A principal $P$ at rate $i$ per period grows as the GP $P, P (1 + i), P (1 + i)^{2}, \dots$ ; annuity payments sum via equation [3].
Radioactive decay & half-lives. Each half-life multiplies the remaining mass by $r = \frac{1}{2}$ : a GP with $∣ r ∣ < 1$ .
Population & epidemic modelling. Early-stage spread with reproduction number $R_{0}$ grows geometrically, term ratio $r = R_{0}$ .
Drug dosing (pharmacokinetics). Concentration after repeated equal doses with constant elimination fraction forms a GP whose steady-state level is the infinite sum $\frac{a}{1 - r}$ .
Digital signal attenuation & reverberation. Echoes decaying by a fixed reflection coefficient per bounce produce a geometric series of amplitudes.
Computer science — algorithm analysis. Costs across the levels of a binary recursion tree ( $1 + 2 + 4 + \dots$ ) are geometric sums central to the Master Theorem.
Fractals & self-similarity. The total length/area of constructions like the Koch curve or Sierpiński triangle is computed as an infinite geometric series.

Historical Development

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 $1 + \frac{1}{4} + \frac{1}{16} + \dots = \frac{4}{3}$ 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 $\frac{1}{2} + \frac{1}{4} + \dots$ , 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 ( $∣ r ∣ < 1$ ).

Subtleties & Limits

The clean dichotomy " $∣ r ∣ < 1$ converges, otherwise diverges" hides a rich frontier. Divergent series like $1 + 2 + 4 + \dots$ 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 $1 + 2 + 3 + \dots = - \frac{1}{12}$ ). 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. $r = - 1$ : $1 - 1 + 1 - 1 + \dots$ , 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 $r \in C$ opens the door to power series, radii of convergence, and the analytic functions of complex analysis.