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 | $$un | $$5×2n−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:
$$un=u1rn−1
For any geometric sequence with starting value $$a and common ratio $$r, we can express it in the form:
$$tn=arn−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:
$$u8 | $$= | $$15(2)7 |
$$= | $$15×128 | |
$$= | $$1920 |
Study the pattern for the following geometric sequence, and write down the next two terms.
$$3, $$15, $$75, $$, $$
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.
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.