# Geometric Sequences

Suppose you were to draw an equilateral triangle on a sheet of paper. It might look something like this:

Now suppose that you draw lines connecting the midpoints of each of the edges of this triangle. This will dissect the larger triangle into four smaller triangles, each of which are equilateral. Three of these smaller triangles will be oriented in the same direction as the original triangle, whereas one will not. Consider the second image below, with the three triangles with the same orientation as the original triangle numbered.

We can continue to draw lines connecting the midpoints of the edges of the marked triangles and counting the resulting triangles that have the same orientation as the original triangle and we see that a pattern emerges.

What one notices is that each time we draw a new triangle by connecting the midpoints of the marked edges, we wind up with three times the number of triangles that were in the previous picture. So (assuming we had enough space) we could draw out the figure that would be the result of doing any number of these dissections. However, if we are only interested in knowing the number of triangles that each image will contain, we can take advantage of the fact that this pattern represents a geometric sequence.

A geometric sequence is a sequence with an initial term, a1 and a common ration, r, where each term after the initial term is obtained by multiplying the previous term by the ratio (a1 cannot be zero, and r cannot be zero or one).

In a geometric sequence, if we know the first term and the ratio, we can determine the nth term by the formula

an = a1*rn – 1

Similarly, if we know the first term and the ratio, we can determine the sum of the first n terms in a geometric sequence by the formula:

Sn =
 a1(1 – rn) 1 – r

For the previous example with the triangles pointed in the same direction, we can show the results in the following table:

 Drawing Number Number of Triangles ratio sum number sum value a1 1 3 S1 1 a2 3 3 S2 4 a3 9 3 S3 13 a4 27 3 S4 40 a5 81 3 S5 121

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# Arithmetic Sequences

At a previous job of mine, there was a policy of holding a dinner party for the company each time we hired a new employee. At these dinners, each employee was treated to a \$20 dinner at the expense of the company. There was also a manager responsible for keeping track of the costs of these dinners.

In computing the costs, the manager noticed that each time there is a new dinner, it was \$20 more expensive than the last one. So if we let a1 represent the cost of the first dinner, and let ai represent the cost of the ith dinner, then we see that ai = ai-1 + 20. Sequences like this, where t arise quite often in practice and are called arithmetic sequences. An arithmetic sequence is a list of numbers where the difference between any two consecutive numbers is constant.

For the example above, the term an will represent the cost of dinner after the nth employee has joined the company (assuming that no employees have left the company over this time period). Also the term Sn will represent the total cost the company has paid towards these dinners.

Before we continue with this example, consider the following table which lists the first five terms of an arithmetic sequence as well as the common difference and the first five sums of this sequence.

 term number term value diff sum number sum value a1 4 3 S1 4 a2 7 3 S2 11 a3 10 3 S3 21 a4 13 3 S4 34 a5 16 3 S5 50

One of the beauties of arithmetic sequences is that if we know the first term (a1) and the common difference (d), then we can easily calculate the terms an and Sn for any n with the following formulas:

an = a1 + d*(n – 1), where d is the common difference.
Sn = n*(a1 + an)/2

We can use these formulas to derive more information about the sequence. For example, if my manager wanted to estimate the cost of dinners once we had added 30 new employees, this would be term a30 of the sequence, which we can evaluate with the above formula by a30 = a1 + d*(n – 1) = 0 + 20*(30 – 1) = 0 + 20 * 29 = 580.

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