Module 2 - Finding the General Term
TL;DR
A geometric sequence is one where you multiply by a common ratio from term to term.
If you divide by \(3\), then the common ratio is \(\frac{1}{3}\).
The \(n\)th term of a geometric sequence is \(a\times r^{n-1}\) where \(a\) is the first term and \(r\) is the common ratio.
Geometric Sequences are those that are multiplied or divided by the same amount each time.
For example:
\(1, 2, 4, 8, 16, …\) is a geometric sequence because you multiply by \(2\) each time;
\(4, 12, 36, 108, 324, …\) is a geometric sequence because you multiply by \(3\) each time;
\(2, -10, 50, -250, 1250, …\) is a geometric sequence because you multiply by \(-5\) each time;
\(81, 27, 9, 3, 1,\frac{1}{3}, …\) is a geometric sequence because you divide by \(3\) each time;
\(20000, 2000, 200, 20, 2, 0.2, …\) is a geometric sequence because you divide by \(10\) each time.
The number that the sequence is multiplied by is called the common ratio. If the sequence is divided by a number, then the common ratio will be less than \(1\).
We can calculate the common ratio for any geometric sequence by dividing the second term by the first term.
For example:
Find the common ratio for the geometric sequence \(3, 6, 12, 24, 48, …\).
The common ratio is \(6\div 3=2\).
Find the common ratio for the geometric sequence \(-2, 6, -18, 54, -162, …\).
The common ratio is \(6\div -2=-3\).
Find the common ratio for the geometric sequence \(324, 108, 36, 12, 4, …\).
The common ratio is \(108\div 324=\frac{1}{3}\).
Find the common ratio for the geometric sequence \(6, 9, 13.5, 20.25, 30.375, …\).
The common ratio is \(9\div 6=\frac{3}{2}\).
For any geometric sequence the \(n\)th term (or general term) is always of the form \(a\times r^{n-1}\).
In this form, \(a\) is the first term and \(r\) is the common ratio.
For example:
Find the \(n\)th term for the geometric sequence \(3, 6, 12, 24, 48, …\).
The common ratio is \(6\div 3=2\).
The \(n\)th term is \(3\times 2^{n-1}\).
Find the \(n\)th term for the geometric sequence \(-2, 6, -18, 54, -162, …\).
The common ratio is \(6\div -2=-3\).
The \(n\)th term is \(-2\times \left(-3\right)^{n-1}\).
Find the \(n\)th term for the geometric sequence \(324, 108, 36, 12, 4, …\).
The common ratio is \(108\div 324=\frac{1}{3}\).
The \(n\)th term is \(324\times \left(\frac{1}{3}\right)^{n-1}\).
Find the \(n\)th term for the geometric sequence \(6, 9, 13.5, 20.25, 30.375, …\).
The common ratio is \(9\div 6=\frac{3}{2}\).
The \(n\)th term is \(6\times \left(\frac{3}{2}\right)^{n-1}\).
| Sequence | First Term | Ratio | \(n\)th Term |
|---|---|---|---|
| \( 3, 6, 12, 24, 48, \dots \) | \( 3 \) ? |
\( 2 \) ? |
\( 3 \times 2^{n-1} \) ? |
| \( 5, 15, 45, 135, 405, \dots \) | \( 5 \) ? |
\( 3 \) ? |
\( 5 \times 3^{n-1} \) ? |
| \( 10, 20, 40, 80, 160, \dots \) | \( 10 \) ? |
\( 2 \) ? |
\( 10 \times 2^{n-1} \) ? |
| \( 2, 8, 32, 128, 512, \dots \) | \( 2 \) ? |
\( 4 \) ? |
\( 2 \times 4^{n-1} \) ? |
| \( 1, 10, 100, 1000, 10000, \dots \) | \( 1 \) ? |
\( 10 \) ? |
\( 1 \times 10^{n-1} \) ? |
| \( 100, 50, 25, 12.5, 6.25, \dots \) | \( 100 \) ? |
\( \frac{1}{2} \) ? |
\( 100 \times (\frac{1}{2})^{n-1} \) ? |
| \( 81, 27, 9, 3, 1, \dots \) | \( 81 \) ? |
\( \frac{1}{3} \) ? |
\( 81 \times (\frac{1}{3})^{n-1} \) ? |
| \( 250, 50, 10, 2, 0.4, \dots \) | \( 250 \) ? |
\( \frac{1}{5} \) ? |
\( 250 \times (\frac{1}{5})^{n-1} \) ? |
| \( 12, 18, 27, 40.5, 60.75, \dots \) | \( 12 \) ? |
\( \frac{3}{2} \) ? |
\( 12 \times (\frac{3}{2})^{n-1} \) ? |
| \( 4, 6, 9, 13.5, 20.25, \dots \) | \( 4 \) ? |
\( \frac{3}{2} \) ? |
\( 4 \times (\frac{3}{2})^{n-1} \) ? |
| \( 27, 18, 12, 8, \frac{16}{3}, \dots \) | \( 27 \) ? |
\( \frac{2}{3} \) ? |
\( 27 \times (\frac{2}{3})^{n-1} \) ? |
| \( 64, 48, 36, 27, \frac{81}{4}, \dots \) | \( 64 \) ? |
\( \frac{3}{4} \) ? |
\( 64 \times (\frac{3}{4})^{n-1} \) ? |
| \( 125, 50, 20, 8, 3.2, \dots \) | \( 125 \) ? |
\( \frac{2}{5} \) ? |
\( 125 \times (\frac{2}{5})^{n-1} \) ? |
| \( 4, 10, 25, 62.5, 156.25, \dots \) | \( 4 \) ? |
\( \frac{5}{2} \) ? |
\( 4 \times (\frac{5}{2})^{n-1} \) ? |
| \( 1, \frac{3}{5}, \frac{9}{25}, \frac{27}{125}, \dots \) | \( 1 \) ? |
\( \frac{3}{5} \) ? |
\( 1 \times (\frac{3}{5})^{n-1} \) ? |
| \( 20, 4, 0.8, 0.16, \dots \) | \( 20 \) ? |
\( \frac{1}{5} \) ? |
\( 20 \times (\frac{1}{5})^{n-1} \) ? |
| \( 16, 24, 36, 54, 81, \dots \) | \( 16 \) ? |
\( \frac{3}{2} \) ? |
\( 16 \times (\frac{3}{2})^{n-1} \) ? |
| \( 1000, 100, 10, 1, 0.1, \dots \) | \( 1000 \) ? |
\( \frac{1}{10} \) ? |
\( 1000 \times (\frac{1}{10})^{n-1} \) ? |
| \( 40, 24, 14.4, 8.64, \dots \) | \( 40 \) ? |
\( \frac{3}{5} \) ? |
\( 40 \times (\frac{3}{5})^{n-1} \) ? |
| \( 32, 80, 200, 500, 1250, \dots \) | \( 32 \) ? |
\( \frac{5}{2} \) ? |
\( 32 \times (\frac{5}{2})^{n-1} \) ? |
Geometric \(n\)th Term
Find the general rule (\(n\)th term) for the geometric sequence below.