A sequence in which each term increases or decreases from the last by a constant factor is called a geometric sequence. We refer to the constant factor the terms are changing by as the common ratio, which will result from dividing any two successive terms \left(\dfrac{t_{n+1}}{t_n}\right).
We denote the first term in the sequence by the letter a and the common ratio by the letter r. For example, the sequence 4,8,16,32\ldots is geometric with a=4 and r=2. The sequence 100,-50,25,-12.5,\ldots is geometric with a=100 and r=\dfrac{-1}{2}.
To describe the rule in words we say "next term is r multiplied by previous term". Therefore we can write any geometric sequence as the recurrence relation: t_n=rt_{n-1},t_1=a or alternatively t_{n+1}=rt_n, where t_1=a.
Note that it is also possible to define the initial term as t_0, this is particularly useful in financial applications of sequences.
We can also find an explicit formula in terms of a and r, this is useful for finding the nth term without listing the sequence.
n | t_{n} | \text{Pattern} |
---|---|---|
1 | 5 | 5\times 2^0 |
2 | 10 | 5\times 2^1 |
3 | 20 | 5\times 2^2 |
4 | 40 | 5\times 2^3 |
... | ||
n | t_n | 5\times 2^{n-1} |
For any geometric progression with starting value a and common ratio r has the terms given by: a,ar,ar^2,ar^3, \ldots. We see a similar pattern to our previous table and can write down the formula for the nth term: t_n=ar^{n-1}.
For any geometric sequence with starting value a and common ratio r, we can express it in either of the following two forms:
Recursive form: t_n=rt_{n-1}, where t_1=a or alternatively t_{n+1}=rt_n, where t_1=a
Explicit form: t_n=ar^{n-1}
Study the pattern for the following geometric sequence, and write down the next two terms.4,\, 12, \,36, \,\ldots
In a geometric progression, T_4=32 and T_6=128.
Solve for r, the common ratio in the sequence.
For the case where r=2, solve for a, the first term in the progression.
Consider the sequence in which the first term is positive. Find an expression for T_n, the general nth term of this sequence.
For any geometric sequence with starting value a and common ratio r, we can express it in either of the following two forms:
Recursive form: t_n=rt_{n-1}, where t_1=a or alternatively t_{n+1}=rt_n, where t_1=a
Explicit form: t_n=ar^{n-1}
When given a formula for the nth term we can generate a table of values for the sequence. For example in the sequence given by the formula t_n=12\times \left(1.5\right)^{n-1}, by substituting for n appropriately and using a calculator, we can generate the following table of the first 6 terms of the sequence:
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
T_n | 12 | 18 | 27 | 40.5 | 60.75 | 91.125 |
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.
Sequences of the form t_n=ar^{n-1} are exponential graphs, and where a>0 they 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 happens when 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 (-1<r<0) zig-zag path - alternating between points on the graph f(n)=ar^{n-1} and f(n)=-ar^{n-1}, depending on the power being odd or even.
Consider the geometric progression with starting value 12 and ratio r=-1.5. This is the same as the example in the previous table but the ratio is now negative. The nth term is given by t_n=12 \times (-1.5)^{n-1}, the table will be the same but the sign of the terms will alternate.
The new table becomes:
n | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|
T_n | 12 | -18 | 27 | -40.5 | 60.75 | -91.125 |
Below is the comparison of the graphs of T_n=12 \times 1.5^{n-1} and T_n=12\times (-1.5)^{n-1}. The graph in red illustrates T_n=12 \times 1.5^{n-1} and the zig-zag graph illustrates T_n=12\times (-1.5)^{n-1}.
Based on the graph above, it can be observed that the odd terms of the zig-zag graph coincide with the terms of the first geometric progression.
Try adjusting the values of a and r in the applet below to observe the effect on the plotted points.
If r \gt 0, the points trend upwards along an exponential function. If r \lt 0, the points zig-zag about the x-axis.
If a \gt 0, the plotted points trend upwards. If a \lt 0, the plotted points trend downwards.
The nth term of a geometric progression is given by the equation T_n=2\times 3^{n-1}.
Complete the table of values:
n | 1 | 2 | 3 | 4 | 10 |
---|---|---|---|---|---|
T_n |
What is the common ratio between consecutive terms?
Plot the points in the table that correspond to n=1,\,n=2,\,n=3, and n=4.
If the plots on the graph were joined they would form:
Consider the following graph of the first 4 terms of a sequence.
Write a recursive rule for T_n in terms of T_{n-1}, including the initial term T_1.
The given table of values represents terms in a geometric sequence.
n | 1 | 2 | 3 | 4 |
---|---|---|---|---|
T_n | 7 | -21 | 63 | -189 |
Identify r, the common ratio between consecutive terms.
Write a simplified expression for the general nth term of the sequence, T_n.
Find the 12th term of the sequence.
We can represent geometric sequences in both tables and graphs.
From a graph of points, the sequence can be read from the y-coordinates.
use recursion to generate a geometric sequence
display the terms of a geometric sequence in both tabular and graphical form and demonstrate that geometric sequences can be used to model exponential growth and decay in discrete situations
deduce a rule for the nth term of a particular geometric sequence from the pattern of the terms in the sequence, and use this rule to make predictions
use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate