A sequence where each term is multiplied by the same number, r, to get the next term is a geometric sequence. The number r can be any non-zero real number. In the following example, r is 2, making each term twice as large as the previous.
Similar to other sequences, the first term is usually called a1, the second a2, and so on.
Consider the following geometric sequence. 2, 6, 18, 54, 162, … Determine the common ratio and find the next three terms.
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: 2and6. If we let r be the common ratio we get the equation 2⋅r=6⇔r=3. The common ratio, r, is 3. To find the next terms, we multiply by 3, three times. 162⋅3486⋅31458⋅3=486=1458=4374 In summary, the common ratio is 3, and the next three terms are 486, 1458, and 4374.
All geometric sequences have a common ratio, r. Using the common ratio, together with the value of the first term of the sequence, a1, an explicit rule describing the sequence can be found. By expressing the terms in a geometric sequence using a1 and r, a pattern emerges. Note that r0 is equal to 1, and that r can be written as r1.
n | an | Using a1 and r |
---|---|---|
1 | a1 | a1⋅r0 |
2 | a2 | a1⋅r1 |
3 | a3 | a1⋅r2 |
4 | a4 | a1⋅r3 |
When n increases by 1, the exponent on r increases by 1 as well. Due to this, and that the exponent is 0 when n is 1, the exponent is always 1 less than n. Expressing this in a general form gives the explicit rule.
an=a1⋅rn−1
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.
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. r=9648=0.5 Substituting r=0.5 and a1=96 into the general rule for geometric sequences gives the desired rule. an=a1⋅rn−1⇒an=96⋅0.5n−1 Now, we can find the eighth term in the sequence by substituting n=8 into the rule above.
For a geometric sequence, it is known that the common ratio is positive, and that a2=4anda4=64. Find the explicit rule for the sequence and give its first six terms.
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 a2a4=r2, which we can solve for r.
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. a1=ra2⇒a1=44=1 With a1 and r, we have enough information to state the explicit rule.
The desired explicit rule is an=4n−1. We already know the terms a1,a2, and a4. Let's use the rule to find the remaining three.
The terms a5 and a6 are evaluated similarly.
n | 4n−1 | an |
---|---|---|
3 | 43−1 | 16 |
5 | 45−1 | 256 |
6 | 46−1 | 1024 |
Thus, the first six terms of the sequence are 1,4,16,64,256,and 1024.
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.
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. r=26=3 To write the rule, we can substitute a1=2 and r=3 into the general rule for geometric sequences. an=a1⋅rn−1⇒an=2⋅3n−1 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.