A sequence in which each term increases or decreases from the last by a constant factor is called a geometric sequence. We refer to the constant factor the terms are changing by as the common ratio, which will result from dividing any two successive terms $$(un+1un).

We denote the first term in the sequence by the letter $$u1 , or $$a, and the common ratio by the letter $$r. For example, the sequence $$4,8,16,32… is geometric with $$u1=4 and $$r=2. The sequence $$100,−50,25,−12.5,… is geometric with $$u1=100 and $$r=−12.

To describe the rule in words we say "next term is $$r multiplied by previous term".

We can find an explicit formula in terms of $$u1 and $$r, this is useful for finding the $$nth term without listing the sequence.

Let's look at a table of a concrete example to see the pattern for the explicit formula. For the sequence $$5,10,20,40,…, we have a starting term of $$5 and a common ratio of $$2, that is $$u1=5 and $$r=2. A table of the sequence is show below:

$$`n`	$$`un`	Pattern
$$1	$$5	$$5×20
$$2	$$10	$$5×21
$$3	$$20	$$5×22
$$4	$$40	$$5×23
...
$$`n`	$$`un`	$$5×2`n`−1

The pattern starts to become clear and we could guess that the tenth term is $$u10=5×29 and the one-hundredth term is $$u100=5×299. And following the pattern, the explicit formula for the $$nth term is $$un=5×2n−1.

For any geometric progression with starting value $$u1 and common ratio $$r has the terms given by: $$u1,u1r,u1r2,u1r3,... We see a similar pattern to our previous table and can write down the formula for the $$nth term:

$$un=u1rn−1

Forms of geometric sequences

For any geometric sequence with starting value $$a and common ratio $$r, we can express it in the form:

$$tn=arn−1

Worked examples

Example 1

For the sequence $$15,30,60,120..., find an explicit rule for the $$nth term and hence, find the $$8th term.

Think: Check that the sequence is geometric, is the next term made by multiplying the previous term by a constant factor? Then write down the starting value $$u1 and common ratio $$r and substitute these into the general form: $$un=u1rn−1

Do: Dividing the second term by the first we get, $$u2u1=3015=21. Checking the ratio between the successive pairs we also get $$21. So we have a geometric sequence with: $$u1=15 and $$r=2. The general formula for this sequence is: $$un=15(2)n−1.

Hence, the $$8th term is:

$$`u`8	$$=	$$15(2)7
	$$=	$$15×128
	$$=	$$1920

Practice questions

Question 1

Study the pattern for the following geometric sequence, and write down the next two terms.

$$3, $$15, $$75, $$, $$

Question 2

Consider the first four terms in this geometric sequence: $$−8, $$−16, $$−32, $$−64

If $$Tn is the $$nth term, evaluate $$T2T1.
Evaluate $$T3T2
Evaluate $$T4T3
Hence find the value of $$T5.

Question 3

In a geometric progression, $$T4=54 and $$T6=486.

Solve for $$r, the common ratio in the sequence. Write both solutions on the same line separated by a comma.
For the case where $$r=3, solve for $$a, the first term in the progression.
Consider the sequence in which the first term is positive. Find an expression for $$Tn, the general $$nth term of this sequence.