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 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.
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.
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.
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.
|n||Using a1 and r|
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.
The desired explicit rule is 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.
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.