Module 2 - Finding the General Term
TL;DR
Linear Sequences increase by a common difference (this is negative if they decrease) which stays the same.
The zeroth term is the term that would be before the first term.
The \(n\)th term for a linear sequence is \(an+b\) where \(a\) is the common difference and \(b\) is the zeroth term.
Linear Sequences are those that increase or decrease by the same amount each time.
For example:
\(3, 7, 11, 15, 19, …\) is a linear sequence because it increases by \(4\) each time;
\(22, 32, 42, 52, 62, …\) is a linear sequence because it increases by \(10\) each time;
\(55, 50, 45, 40, 35, …\) is a linear sequence because it decreases by \(5\) each time;
\(0.7, 0.9, 1.1, 1.3, 1.5, …\) is a linear sequence because it increases by \(0.2\) each time;
\(3000, 2100, 1200, 300, -600, …\) is a linear sequence because it decreases by \(900\) each time;
\(7.5, 6, 4.5, 3, 1.5, 0, …\) is a linear sequence because it decreases by \(1.5\) each time.
The number that the sequence increases by is called the common difference. If the sequence decreases, then the common difference will be negative.
We can calculate the common difference for any linear sequence by subtracting the first term from the second term.
For example:
Find the common difference for the linear sequence \(7, 13, 19, 25, 31, …\).
The common difference is \(13-7=6\).
Find the common difference for the linear sequence \(25, 22, 19, 16, 13, …\).
The common difference is \(22-25=-3\).
Find the common difference for the linear sequence \(0.8, 1.9, 3, 4.1, 5.2, …\).
The common difference is \(1.9-0.8=1.1\).
The zeroth term of a sequence is the term that would be before the first term.
We can calculate the zeroth term by subtracting the common difference from the first term.
For example:
Find the zeroth term for the linear sequence \(7, 13, 19, 25, 31, …\).
The common difference is \(13-7=6\).
The zeroth term is \(7-6=1\).
Find the zeroth term for the linear sequence \(25, 22, 19, 16, 13, …\).
The common difference is \(22-25=-3\).
The zeroth term is \(25-\left(-3\right)=28\).
Find the zeroth term for the linear sequence \(0.8, 1.9, 3, 4.1, 5.2, …\).
The common difference is \(1.9-0.8=1.1\).
The zeroth term is \(0.8-1.1=-0.3\).
For any linear sequence the \(n\)th term (or general term) is always of the form \(an+b\).
In this form, \(a\) is the common difference and \(b\) is the zeroth term.
For example:
Find the \(n\)th term for the linear sequence \(7, 13, 19, 25, 31, …\).
The common difference is \(13-7=6\).
The zeroth term is \(7-6=1\).
The \(n\)th term is \(6n+1\).
Find the \(n\)th term for the linear sequence \(25, 22, 19, 16, 13, …\).
The common difference is \(22-25=-3\).
The zeroth term is \(25-\left(-3\right)=28\).
The \(n\)th term is \(-3n+28\).
Find the \(n\)th term for the linear sequence \(0.8, 1.9, 3, 4.1, 5.2, …\).
The common difference is \(1.9-0.8=1.1\).
The zeroth term is \(0.8-1.1=-0.3\).
The \(n\)th term is \(1.1n+-0.3=1.1n-0.3\).
| Sequence | Difference | Zero Term | \(n\)th Term |
|---|---|---|---|
| \( 4, 7, 10, 13, \dots \) | \( +3 \) ? |
\( 1 \) ? |
\( 3n + 1 \) ? |
| \( 5, 10, 15, 20, \dots \) | \( +5 \) ? |
\( 0 \) ? |
\( 5n \) ? |
| \( 10, 8, 6, 4, \dots \) | \( -2 \) ? |
\( 12 \) ? |
\( -2n + 12 \) ? |
| \( 1, 5, 9, 13, \dots \) | \( +4 \) ? |
\( -3 \) ? |
\( 4n - 3 \) ? |
| \( 20, 13, 6, -1, \dots \) | \( -7 \) ? |
\( 27 \) ? |
\( -7n + 27 \) ? |
| \( 0.5, 1.5, 2.5, 3.5, \dots \) | \( +1 \) ? |
\( -0.5 \) ? |
\( n - 0.5 \) ? |
| \( -3, 3, 9, 15, \dots \) | \( +6 \) ? |
\( -9 \) ? |
\( 6n - 9 \) ? |
| \( 100, 90, 80, 70, \dots \) | \( -10 \) ? |
\( 110 \) ? |
\( -10n + 110 \) ? |
| \( 7, 8, 9, 10, \dots \) | \( +1 \) ? |
\( 6 \) ? |
\( n + 6 \) ? |
| \( -2, -5, -8, -11, \dots \) | \( -3 \) ? |
\( 1 \) ? |
\( -3n + 1 \) ? |
| \( 11, 22, 33, 44, \dots \) | \( +11 \) ? |
\( 0 \) ? |
\( 11n \) ? |
| \( 15, 11, 7, 3, \dots \) | \( -4 \) ? |
\( 19 \) ? |
\( -4n + 19 \) ? |
| \( 2, 10, 18, 26, \dots \) | \( +8 \) ? |
\( -6 \) ? |
\( 8n - 6 \) ? |
| \( -10, -5, 0, 5, \dots \) | \( +5 \) ? |
\( -15 \) ? |
\( 5n - 15 \) ? |
| \( 45, 36, 27, 18, \dots \) | \( -9 \) ? |
\( 54 \) ? |
\( -9n + 54 \) ? |
| \( 1.2, 1.4, 1.6, 1.8, \dots \) | \( +0.2 \) ? |
\( 1.0 \) ? |
\( 0.2n + 1 \) ? |
| \( -1, -2, -3, -4, \dots \) | \( -1 \) ? |
\( 0 \) ? |
\( -n \) ? |
| \( 6, 13, 20, 27, \dots \) | \( +7 \) ? |
\( -1 \) ? |
\( 7n - 1 \) ? |
| \( 50, 48.5, 47, 45.5, \dots \) | \( -1.5 \) ? |
\( 51.5 \) ? |
\( -1.5n + 51.5 \) ? |
| \( 3, 3.5, 4, 4.5, \dots \) | \( +0.5 \) ? |
\( 2.5 \) ? |
\( 0.5n + 2.5 \) ? |
\(n\)th Term Practice
Find the general rule (the \(n\)th term) for the sequence below.