# Geometric Sequence — Formula, Sum, Examples

### TL;DR

A geometric sequence is a list of numbers where every term after the first is found by multiplying the previous term by a fixed common ratio r. This article covers the nth-term and sum formulas, three worked examples at Quick/Standard/Stretch tiers, the role of |r|<1 for convergent infinite series, and the difference between a sequence and a series.

### 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 a₁,a₂,a₃,… 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**, a₁ = a is the **first term**, and the general nth term is

\[
a_n = a \cdot r^{n - 1}.
\]

Examples — 5,10,20,40,… (a=5, r=2). ; 81,27,9,3,1,… (a=81, r=1/3). ; 1,−2,4,−8,16,… (a=1, r=−2).

> **Quick facts.**  
> - **nth term:** \(a_n = a \cdot r^{n-1}\)  
> - **Common ratio:** \(r = \frac{a_{n+1}}{a_n}\), the same for every consecutive pair.  
> - **Finite sum:** \(S_n = \frac{a(1 - r^n)}{1 - r}\) for \(r \neq 1\).  
> - **Infinite sum:** \(S_{\infty} = \frac{a}{1 - r}\) 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|≥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,…  
**Final answer:** \(a_8 = 256\).

**Standard (Wrong Path First — Watch How This Goes Wrong).** Find the sum 1+1/2+1/4+1/8+⋯  
**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.
**Final answer:** a=2, r=3. The sequence is 2,6,18,54,162,486,…

### 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,…
- **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…=0.3+0.03+0.003+⋯ is a geometric series summing to 1/3.
- **Population growth.** Before resource limits hit, populations grow geometrically (the early phase of exponential growth).

### The Geometric Sequence Errors That Cost Most Marks

1. **Using the arithmetic-sum formula on a geometric sequence.** Use \(S_n = a(1 - r^n)/(1 - r)\).
2. **Assuming all infinite sums diverge.** Check |r|. If |r|<1, the infinite sum converges.
3. **Confusing the first term with the second term.** Use the first term as a.
4. **Forgetting the −1 in the exponent.** Use \(a_n = a \cdot r^{n-1}\).

### The Mathematicians Who Shaped Sequences

**Pingala (c. 200 BCE, India)** worked with geometric and arithmetic sequences in his *Chandaḥśāstra*.
**Archimedes (c. 287–212 BCE, Greece)** computed the infinite sum 1/4+1/16+1/64+⋯=1/3.
**Fibonacci (c. 1170–1250, Italy)** introduced sequences to medieval Europe in *Liber Abaci* (1202).

### Conclusion

- A **geometric sequence** has a constant ratio r between consecutive terms.
- The nth 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 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

1. Find the 10th term of 5,15,45,135.…
2. Find the sum of the first 6 terms of 4,8,16,32.…
3. Compute 1+1/3+1/9+1/27+⋯ (infinitely many terms).

### Frequently Asked Questions

**What is the difference between a geometric sequence and an arithmetic sequence?**  
An arithmetic sequence has a constant difference between terms. A geometric sequence has a constant ratio between terms.

**What is the common ratio?**  
The fixed number r that multiplies one term to produce the next.

**When does the infinite sum exist?**  
When |r|<1.

**Can the common ratio be negative?**  
Yes. r=−2 gives 1,−2,4,−8,….

**Can r=0 work?**  
No.

**What is a geometric series?**  
The sum of the terms of a geometric sequence.