5.05 Geometric sequences

A sequence in which each term increases or decreases from the last by multiplying by a constant factor is called a geometric sequence. The constant factor multiplying each term in the sequence to create the next term is referred to as the common ratio, which will result from dividing any two successive terms \dfrac{u_{n+1}}{u_n}.

The first term in the sequence is denoted by the letter a and the common ratio is denoted by R. For example, the sequence 4, 8, 16, 32 is geometric with a = 4 and R = 2. The sequence 100, -50, 25, -12.5 is geometric with a = 100 and R =-\dfrac{1}{2}.

Since, u_2=R\times u_1, u_3=R\times u_2, and so on, any geometric sequence can be written as a recurrence relation, which was found  previously in this chapter  :

u_{n+1}=Ru_n, u_0=a

An explicit generating rule can be found in terms of a and R. This is useful for finding the nth term without listing the sequence or having to use the previous term in the sequence each time to find the next term.

Consider the following table to see the pattern for the explicit formula. For the sequence 5,10,20,40, \ldots, the starting term is 5 and there is a common ratio of 2, that is a=5 and R=2.

A table of the sequence is show below:

By correctly identifying the pattern, the tenth term becomes u_{10}=10\times 2^9 and the one-hundredth term would be u_{100}=5\times 2^{99}. And following the pattern, the explicit formula for the nth term is u_n=5\times 2^{n-1}.

For any geometric progression with starting value a and common ratio R and with terms given by: a,aR,aR^2,aR^3,... , a similar pattern can be observed. Hence, the explicit generating rule for the nth term in any geometric sequence is given by:

u_n=aR^{n-1}

Examples

When given a formula for the nth term, a table of values can be generated for the sequence. For example, in the sequence given by the formula u_n=12\times \left(1.5\right)^{n - 1}, by substituting for n appropriately and using a calculator, the following table can be created, listing the first 6 terms of the sequence:

Perhaps more interesting though is the different types of graphs that geometric sequences correspond to. The graphs are not linear like arithmetic progressions, except for the trivial case of R = 1. The path of points plotted from a geometric sequence follow an exponential curve for positive values of R (graph can be seen below in the next worked example).

Geometric sequences of the form u_n=aR^{n-1}, where a>0 will follow:

• The path of an exponential growth function for R>1.

• The path of an exponential decay function for 0<R<1.

• If a is negative the path will be reflected about the x-axis.

What if R is negative? The values of successive terms flip their sign so that the graph is depicted as either a growing (|R|>1) or diminishing (|R|<1) zig-zag path - alternating between points on the graph f(n)=a|R|^{n-1} and f(n)=-a|R|^{n-1}, depending on the power being odd or even.

For the geometric progression with starting value 12 and ratio R = -1.5, we can create a table and plot a graph of the sequence.

This is the same as the example in the previous table but the ratio is now negative. The nth term is given by u_n=12\times \left(-1.5\right)^{n-1}, the table will be the same but the sign of the terms will alternate.

The new table becomes:

Checking, for n = 1, we have u_1=12\times \left(-1.5\right)^{1-1}=12 and for n = 2 we have u_2=12\times \left(-1.5\right)^{2-1}=-18, continuing on even numbered terms become negative and odd numbered terms become positive.

Here is a graph of the two geometric sequences depicted in both tables. Note that the odd terms of the zig-zag graph coincide with the terms of the first geometric progression.

Examples