Sequences - I - The Geometric Sequence

This section will provide an introduction to geometric sequences.

A geometic sequence is a sequence where we obtain each term after the first by multiplying the previous term by a common multiplier (frequently called the common ratio). We find the common ratio of a sequence by dividing any term (other than the first) by the preceeding term. For example we might have the sequence 3, 6, 12, 24, 48. .... If we divide 48 by 24 we get the common ratio = 48/24 = 2. Dividing 12/6 gives the same result. In many texts the common ratio is referred to by the letter r. The terms of the sequence are represented by a1, a2, a3 ... an. For every positive integer n we can define a term in the sequence by the relationship an+1 = r*al. The general terms of a geometric sequence can we written as:

Individual Terms in a Sequence

To recap: When sequences are talked about in a general way, with symbols the terms of the sequence are describes as a1, a2... The ratio of sequential terms, the common ratio is usually represented by the letter r. So you can say that r = an+1/an. In other words r is equal to the ratio of adjacent terms in the sequence. Lets take a sequence and write some of the terms.

Let's use what was demonstrated above to solve a problem. Suppose you are given the geometric sequence 4, 12, 36, 108, ... and asked to find the expression for the general term of this sequence.

To help solidify this let's look at a few problems. We'll start simply and progress to the (impossibly) difficult.

Determining the type of sequence and the general expression when given a sequence



Sum of the Terms in a Geometric Sequence

Up to this point we have discussed the individual terms of a geometric sequence. The next question to address is the summing of all those individual terms to get the sum of the terms in a sequence. In the case of an arithmetic sequence we added a constant amount to a term to get the next term. For a geometric sequence we multiply a term by a constant to get the next term. Let's use that information to see what we can learn about sums of geometric sequences.

Lets next consider an geometric sequence such as 1, 2, 4, 8, 16, 32

To start let's look at a problem that asks us to sum the first 7 terms of the geometric sequence ...2, 6, 18, 54, ...

  • ..

    One question we can ask about geometric sequences is, "What happens to the sum of the terms as n approaches infinity?"



    ====================================================================

    Determining the sum of the terms of a sequence



    Word Problems

    We'll end this page with a couple of word problems. To help you work these let me tell you a little bit about what is usually called exponential growth. Assume you invest $1000 in a mutual fund that pays 8% per year (long term average). Here is how the amount of money you have will change:

    Now let's look at a couple of word problems that involve investments.