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Sequences and Series

Describing Geometric Sequences

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Concept

Geometric Sequence

A geometric sequence is a sequence in which the ratio r between consecutive terms is a nonzero constant. This ratio is called the common ratio. The following is an example geometric sequence with a first term 3 and common ratio 2.

The geometric Sequence: 3,6,12,24,48,... has a common ration of 2

The behavior of geometric sequences generally depend on the values of the first term and the common ratio. The following table shows the effects of these parameters.

a1>0 a1<0
r>1 Increasing
Decreasing
r=1 Constant

Constant

0<r<1 Decreasing
Increasing
r<0 Alternating

Alternating

Like for any other sequence, the first term of a geometric sequence is denoted by a1, the second by a2, and so on. Since geometric sequences have a common ratio r, once one term is known, the following term can always be obtained by multiplying the known term by r.

General geometric sequence

In particular, if just the first term a1 is known, all the following terms can be found by multiplying by r a specific number of times. Therefore, geometric sequences have the following general form.

Example

Find the common ratio

fullscreen
Consider the following geometric sequence.
Determine the common ratio and find the next three terms.
Show Solution expand_more
In geometric sequences, the terms increase or decrease by a common ratio. Since we know that this sequence is geometric, it's enough to find the ratio between two consecutive terms. The ratio for the others must then be the same. Let's take the first two:
If we let r be the common ratio we get the equation
The common ratio, r, is 3. To find the next terms, we multiply by 3, three times.
In summary, the common ratio is 3, and the next three terms are 486, 1458, and 4374.

Rule

Explicit Rule of Geometric Sequences

Every geometric sequence can be described by a function known as the explicit rule, which receives as input the position of a term n and returns as output the term's value An explicit rule for a geometric sequence has the following general form.

Here, a1 is the first term of the sequence and r is the common ratio.

Proof

Proof by Induction
Note that every geometric sequence has a common ratio r. Therefore, it is possible to obtain every term of the sequence by multiplying the first term a1 by this common ratio enough times. Therefore, knowing a1 and r is enough to describe the whole geometric sequence.
Interactive applet showing how to rewrite the first five terms of a geometric sequence as an expression involving just the common difference and the first term
It is easier to identify a pattern that can be used to write a general expression for the explicit rule by making a table. Note that by the Zero Exponent Property, r0 is equal to 1. Furthermore, r can be written as r1.
n Using a1 and r
1 a1 a1r0
2 a2 a1r1
3 a3 a1r2
4 a4 a1r3

It can be seen that the exponent of the common ratio is always 1 less than the value of the position n. This allows to write the explicit rule in the same form as the formula given at the beginning.

Example

Find and use the explicit rule of the geometric sequence

fullscreen
The first four terms of a geometric sequence are
96,48,24,and 12.
Find the explicit rule describing the geometric sequence. Then, use the rule to find the eighth term of the sequence.
Show Solution expand_more
To write the explicit rule for the sequence, we first have to find the common ratio, r. To do so, we can divide any term in the sequence by the term that precedes it. Let's use the second and first term.
Substituting r=0.5 and a1=96 into the general rule for geometric sequences gives the desired rule.
Now, we can find the eighth term in the sequence by substituting n=8 into the rule above.
a8=0.75
The eighth term in the sequence is 0.75.

Example

Use the terms to find the rule of the geometric sequence

fullscreen
For a geometric sequence, it is known that the common ratio is positive, and that
Find the explicit rule for the sequence and give its first six terms.
Show Solution expand_more

The terms we've been given are not consecutive. Therefore, we can't directly find r. However, the terms a2 and a4 are 2 positions apart, so the ratio between them must be r2.

This gives the equation
which we can solve for r.
16=r2
r2=16
r=±4

r=4
Now that we know the common ratio, we have to find a1 as well, to be able to write the explicit rule. Knowing one term, a subsequent one can by found by multiplying by r. Therefore, a previous term is instead found by dividing by r. Using a2 and r this way, we can find a1.
With a1 and r, we have enough information to state the explicit rule.

The desired explicit rule is We already know the terms a1,a2, and a4. Let's use the rule to find the remaining three.

a3=42
a3=16

The terms a5 and a6 are evaluated similarly.

n
3 16
5 256
6 1024
Thus, the first six terms of the sequence are
1,4,16,64,256,and 1024.

Example

Model the situation using a geometric sequence

fullscreen

Pelle's good friend, Lisa, decides to play a trick on Pelle. While he is away, she rearranges his pellets so that they are grouped in a geometric sequence instead of an arithmetic one. The first group has 2 pellets, the second has 6, the third has 18, and so on. Find a rule describing this sequence. After finishing the seventh group, Lisa counted 3273 remaining pellets. Use the rule to figure out whether there are enough to make an eighth group.

Show Solution expand_more
To begin, we'll write the explicit rule describing this particular geometric sequence. It is given that a1=2. To find the common ratio, r, we can divide the second term by the first.
To write the rule, we can substitute a1=2 and r=3 into the general rule for geometric sequences.
To find if there are enough pellets to finish the eighth group, we must know the eighth term in the sequence. We'll substitute n=8 into the rule.
The eighth term, is greater than the number of remaining pellets, 3273. Thus, there are not enough pellets for Lisa to make the eighth group.
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