Nth Term of a GP — Formula and Worked Examples

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 family, where you add a fixed amount instead of multiplying. It is also covered, under its other name, in geometric sequence.

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.

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

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.