A List Whose Next Term Is Always a Multiplication Away
Some lists of numbers grow by adding a fixed amount. Others grow by multiplying by a fixed amount. The second kind — the geometric sequence — is what compound interest looks like, what radioactive decay looks like, and what a viral video's view count looks like in its first 24 hours.
A geometric sequence is one of the smallest mathematical objects that captures the essence of exponential change. Master it once and the same pattern appears in twenty later topics.
What a Geometric Sequence Is
A geometric sequence is an ordered list of numbers $a_1, a_2, a_3, \ldots$ such that the ratio between any two consecutive terms is the same constant $r$:
$$\frac{a_{n+1}}{a_n} = r \quad \text{for every } n.$$
$r$ is the common ratio, $a_1 = a$ is the first term, and the general $n$th term is
$$a_n = a \cdot r^{n - 1}.$$
Examples — $5, 10, 20, 40, \ldots$ ($a = 5$, $r = 2$). $;;81, 27, 9, 3, 1, \ldots$ ($a = 81$, $r = 1/3$). $;;1, -2, 4, -8, 16, \ldots$ ($a = 1$, $r = -2$).
Quick facts.
$n$th term: $a_n = a \cdot r^{n-1}$.
Common ratio: $r = a_{n+1} / a_n$, the same for every consecutive pair.
Finite sum: $S_n = a(1 - r^n)/(1 - r)$ for $r \neq 1$.
Infinite sum: $S_{\infty} = a/(1 - r)$ when $|r| < 1$ (the series converges).
Sequence vs series: a sequence is a list of terms; a series is the sum of a sequence's terms.
Grade introduced: CBSE Class 11 (sequences chapter); CCSS-M HSF-LE.A.2 (construct linear and exponential functions, including geometric sequences); NCERT Class 11 Chapter 9 — Sequences and Series.
The Sum of a Geometric Sequence
When the terms of a geometric sequence are added, the result is called a geometric series. The first $n$ terms sum to
$$S_n = \frac{a(1 - r^n)}{1 - r}, \quad r \neq 1.$$
When $r = 1$, every term equals $a$ and $S_n = na$.
When $|r| < 1$ and we sum infinitely many terms, the $r^n$ factor shrinks to zero, and the infinite sum becomes
$$S_{\infty} = \frac{a}{1 - r}.$$
When $|r| \geq 1$, the terms do not shrink and the infinite sum diverges (does not exist as a finite number).
Examples of Geometric Sequence
Quick. Find the 8th term of the sequence $2, 4, 8, 16, \ldots$.
$a = 2$, $r = 2$, $n = 8$.
$$a_8 = 2 \cdot 2^{7} = 2 \cdot 128 = 256.$$
Final answer: $a_8 = 256$.
Standard (Wrong Path First — Watch How This Goes Wrong). Find the sum $1 + \tfrac{1}{2} + \tfrac{1}{4} + \tfrac{1}{8} + \cdots$ (infinitely many terms).
The wrong path. The rusher thinks "infinitely many terms must add to infinity" and reports $\infty$.
The flaw: the terms shrink toward zero. Adding shrinking terms can give a finite total. The geometric series with $|r| < 1$ is exactly this case.
The rescue. Here $a = 1$, $r = 1/2$, and $|r| < 1$, so the infinite sum converges.
$$S_{\infty} = \frac{a}{1 - r} = \frac{1}{1 - 1/2} = \frac{1}{1/2} = 2.$$
Final answer: the infinite sum is exactly $2$.
Stretch. The third term of a geometric sequence is 18 and the sixth term is 486. Find the first term and the common ratio.
Use $a_3 = ar^2 = 18$ and $a_6 = ar^5 = 486$.
Divide: $\dfrac{a_6}{a_3} = \dfrac{ar^5}{ar^2} = r^3 = \dfrac{486}{18} = 27$.
So $r = 3$.
Substitute back: $a \cdot 3^2 = 18 \implies a = 2$.
Final answer: $a = 2$, $r = 3$. The sequence is $2, 6, 18, 54, 162, 486, \ldots$.
Where Geometric Sequences Show Up
The geometric sequence is one of the most common patterns in nature, finance, and engineering.
Compound interest. A principal $P$ at rate $r%$ compounded annually produces the geometric sequence $P, P(1 + r), P(1 + r)^2, \ldots$.
Radioactive decay. After each half-life, the remaining quantity is half the previous — geometric with $r = 1/2$.
Bouncing ball. A ball that bounces to 80% of its previous height produces a geometric sequence of peak heights with $r = 0.8$.
Repeating decimals. $0.333\ldots = 0.3 + 0.03 + 0.003 + \cdots$ is a geometric series summing to $1/3$.
Population growth. Before resource limits hit, populations grow geometrically (the early phase of exponential growth).
The destination, in every direction: any quantity that multiplies by a fixed factor at each step is a geometric sequence.
The Geometric Sequence Errors That Cost Most Marks
1. Using the arithmetic-sum formula on a geometric sequence
Where it slips in: Asked to sum a geometric sequence, the student uses $S = (n/2)(a + a_n)$.
Don't do this: Apply the AP sum formula to GP terms.
The correct way: Use $S_n = a(1 - r^n)/(1 - r)$.
2. Assuming all infinite sums diverge
Where it slips in: $1 + 1/2 + 1/4 + \cdots$ — student declares the sum infinite.
Don't do this: Treat any infinite series as divergent.
The correct way: Check $|r|$. If $|r| < 1$, the infinite sum converges to $a/(1 - r)$.
3. Confusing the first term with the second term
Where it slips in: The sequence $2, 4, 8, 16, \ldots$ has $a_1 = 2$, but the student substitutes $a = 4$.
Don't do this: Use the second term as $a$.
The correct way: $a = a_1$ is the very first term — the term before any multiplication by $r$.
4. Forgetting the $-1$ in the exponent
Where it slips in: Writing $a_n = a \cdot r^n$ instead of $a_n = a \cdot r^{n-1}$.
Don't do this: Treat the first term as $a \cdot r$.
The correct way: $a_1 = a \cdot r^0 = a$. The exponent $n - 1$ ensures the first term is the unmultiplied $a$.
The real-world version. In March 2020, the early phase of the COVID-19 pandemic was a geometric sequence in case count, with a doubling time of about 3 days — common ratio approximately 1.26 per day. Governments that responded to the first hundred cases bought 10 days of preparation; those that waited for the first thousand cases bought less than three days.
The geometric sequence rewards early action because the next term is always a fixed factor of the current one — the longer you wait, the steeper the climb you're starting from.
The Mathematicians Who Shaped Sequences
Pingala (c. 200 BCE, India) worked with geometric and arithmetic sequences in his Chandaḥśāstra, originally for poetic meter.
Archimedes (c. 287–212 BCE, Greece) computed the infinite sum $1/4 + 1/16 + 1/64 + \cdots = 1/3$ using the method of exhaustion — the first known rigorous infinite geometric series.
Fibonacci (c. 1170–1250, Italy) introduced sequences to medieval Europe in Liber Abaci (1202), opening the door to European understanding of geometric growth.
Conclusion
A geometric sequence has a constant ratio $r$ between consecutive terms.
The $n$th term is $a_n = a r^{n-1}$.
The finite sum is $S_n = a(1 - r^n)/(1 - r)$; the infinite sum is $a/(1 - r)$ when $|r| < 1$.
The single most common mistake is using the arithmetic-sum formula on a geometric sequence.
Geometric sequences model compound interest, radioactive decay, and the early phase of any exponential growth.
Try It Yourself — Three Problems
Find the 10th term of $5, 15, 45, 135, \ldots$.
Find the sum of the first 6 terms of $4, 8, 16, 32, \ldots$.
Compute $1 + \tfrac{1}{3} + \tfrac{1}{9} + \tfrac{1}{27} + \cdots$ (infinitely many terms).
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