# Nth Term of a GP — Formula and Worked Examples

**TL;DR**  
The nth term of a GP (geometric progression) is found with \( a_n = ar^{n-1} \), where \( a \) is the first term and \( r \) is the common ratio. This article gives you the formula, a clean derivation, the term-from-the-end version, six worked examples, and the slips that cost marks — so you can find any term without writing out the whole sequence.

## What is the nth Term of a GP?

The **nth term of a GP** is the value of the term in position \( n \) of a geometric progression, given by the formula \( a_n = ar^{n-1} \). It lets you jump straight to any term without listing every term before it.

First, the terms in that formula. A **geometric progression** is a sequence where each term is the previous one multiplied by a fixed number, the **common ratio** \( r \). In \( 3, 6, 12, 24, \dots \), each term is double the last, so \( r=2 \) and the first term \( a=3 \). You find \( r \) by dividing any term by the one before it:

\[ r = \frac{a_2}{a_1} = \frac{6}{3} = 2 \]

A general GP is written:
\( a, ar, ar^2, ar^3, \dots \)

This is the multiplicative cousin of the [arithmetic progressions](/content/math/algebra/arithmetic-progressions/index.html) family, where you _add_ a fixed amount instead of multiplying. It is also covered, under its other name, in [geometric sequence](/content/math/algebra/geometric-sequence/index.html).

## The nth-term formula and where it comes from

The formula is:
\[ a_n = ar^{n-1} \]

where \( a \) is the first term, \( r \) is the common ratio, \( n \) is the position of the term you want, and \( a_n \) is its value.

**The derivation is just counting how many times you multiply by \( r \).** To reach the first term you multiply by \( r \) zero times; the second term, once; the \( n \)th term, \( (n-1) \) times:

\[ a_1 = a \]  
\[ a_2 = a \cdot r \]  
\[ a_3 = ar \cdot r \]  
\[ a_n = ar^{n-1} \]

The exponent is \( (n-1) \), not \( n \), because the first term already exists before any multiplication happens. That single fact is where most errors live.

### The nth term from the end

Sometimes you want a term counted from the _last_ term of a finite GP, not the first. If the GP has \( n \) terms and last term \( l \), the \( k \)th term from the end is:

\[ a_{n-k+1} = l \cdot \left(\frac{1}{r}\right)^{k-1} \]

You divide by \( r \) as you walk backwards, which is the same as multiplying by \( \frac{1}{r} \).

## Examples of the Nth Term of a GP

Six problems, easier to harder, all using \( a_n = ar^{n-1} \).

### Example 1

**Find the common ratio of 5, 15, 45, 135,…**

Divide a term by the one before it: \( \frac{15}{5} = 3 \) and \( \frac{45}{15} = 3 \).

Final answer: \( r = 3 \).

### Example 2

**Find the 6th term of 2, 6, 18, 54,…**

Here \( a=2 \), \( r=3 \), \( n=6 \):  
\[ a_6 = ar^{n-1} = 2 \cdot 3^{6-1} = 2 \cdot 3^5 = 486 \]

Final answer: 486.

### Example 3

**Find the 8th term of 1, 2, 4, 8,…**

Here \( a=1 \), \( r=2 \), \( n=8 \):  
\[ a_8 = 1 \cdot 2^{8-1} = 2^7 = 128 \]

Final answer: 128.

### Example 4

**Find the 5th term of a GP with first term 2 and common ratio \( \frac{1}{3} \).**

Here \( a=2 \), \( r=\frac{1}{3} \), \( n=5 \):  
\[ a_5 = 2 \cdot \left(\frac{1}{3}\right)^{5-1} = \frac{2}{81} \]

Final answer: \( \frac{2}{81} \).

### Example 5

**In the GP 3,6,12,…, which term equals 192?**

Here \( a=3 \), \( r=2 \), \( a_n=192 \):  
\[ 192 = 3 \cdot 2^{n-1} \]  
Solve for \( n \):  
\[ n = 7 \]

Final answer: the 7th term.

### Example 6

**A bacterial culture starts with 500 cells and triples every hour. How many cells after 4 hours?**

Here \( a=500 \), \( r=3 \), since the start counts as term 1 (after 4 hours is the 5th term):  
\[ a_5 = 500 \cdot 3^{4} = 40500 \]

Final answer: 40,500 cells.

## Where the nth term of a GP earns its keep

A GP captures anything that grows or shrinks by a constant _factor_ rather than a constant amount, and the nth-term formula is how you predict where it lands.

- **Compound interest:** Money growing at a fixed percentage each period follows a GP; the nth term gives the balance after \( n \) periods without year-by-year addition.
- **Population and decay:** Bacterial growth, radioactive decay, and drug clearance all multiply by a fixed factor each step — the nth term is the prediction tool.
- **Computing and data:** Storage doubling, signal halving, and the depth of binary structures are all geometric.

The systematic study of such ratios runs back to **Euclid**, whose _Elements_ (around 300 BCE) treats geometric progressions formally.

## Common Mistakes With the Nth Term of a GP

### Mistake 1: Using \( r^n \) instead of \( r^{n-1} \)
**Don't do this:** \( a_6 = ar^6 \).  
**The correct way:** \( a_6 = ar^{5} \).

### Mistake 2: Finding the common ratio by subtracting
**Don't do this:** \( r = 12 - 4 \).  
**The correct way:** \( r = \frac{12}{4} = 3 \).

### Mistake 3: Mishandling a fractional or negative ratio
**Don't do this:** Powering only the bottom partially.  
**The correct way:** Power the _whole_ ratio and keep signs correctly.

## Conclusion
- The **nth term of a GP** is \( a_n = ar^{n-1} \).
- The exponent is \( (n-1) \), not \( n \).
- The common ratio is found by _dividing_ consecutive terms.
- A negative ratio alternates signs; a fractional ratio shrinks the terms.

## Frequently Asked Questions

**What is the formula for the nth term of a GP?**  
\( a_n = ar^{n-1} \).

**Can the common ratio be negative or a fraction?**  
Yes to both. A negative ratio makes terms alternate in sign; a fractional ratio makes the terms shrink toward zero.