An arithmetic sequence is not merely a list where “you keep adding the same number”. Its constant difference creates a linear relationship between term number and term value.
This guide develops arithmetic sequences as structure: common difference, nth-term rule, distant terms, reverse problems, arithmetic means and the connection to straight-line graphs.
The defining feature
In an arithmetic sequence, consecutive terms differ by the same constant amount d.
3, 7, 11, 15, ...
common difference d = 4The sequence grows by four each step.
General term
If the first term is a and the common difference is d, then:
Tn = a + (n-1)dThe expression says: begin at the first term, then make n-1 equal jumps of size d.
Worked example 1
Find the 25th term of 6, 10, 14, 18, …
a=6, d=4
T25 = 6 + 24(4)
= 102The 25th term is 102.
Equivalent nth-term form
The same rule can be simplified:
Tn = 6 + 4(n-1)
= 4n + 2Both forms are correct. The first reveals the first-term-and-step structure; the second is compact for algebraic use.
Worked example 2: decreasing arithmetic sequence
For 50, 43, 36, 29, … the common difference is -7.
Tn = 50 + (n-1)(-7)
= 57 - 7nA negative difference simply means the linear relationship slopes downward as n increases.
Reverse problem: which term is this?
In the sequence 4, 9, 14, 19, …, which term is 199?
Tn = 5n - 1
5n - 1 = 199
5n = 200
n = 40199 is the 40th term.
Arithmetic means
If three consecutive terms of an arithmetic sequence are x, y and z, then y sits halfway between x and z:
y-x = z-y
2y = x+z
y = (x+z)/2This is why the middle of three equally spaced values is their arithmetic mean.
Connection to straight-line graphs
Plot term number n horizontally and term value Tn vertically. An arithmetic sequence lies on a straight line because Tn has the form dn+c.
The common difference acts like gradient: each increase of 1 in n changes the term by d.
Common errors
Using n jumps instead of n-1. The first term requires zero jumps from itself.
Losing the sign of a negative difference. Decreasing sequences use negative d.
Assuming a constant ratio is arithmetic. A geometric pattern is different; arithmetic sequences require constant difference.
Diagnostic table
| Observed mistake | Likely issue | Repair |
|---|---|---|
| T2=a+2d | Step count error | Count intervals between positions |
| Ignores negative d | Direction lost | Write consecutive differences explicitly |
| Finds next term only | Local rule | Move to Tn=a+(n-1)d |
Practice
- Find the 30th term of 2, 7, 12, 17, …
- Find the nth term of 40, 34, 28, 22, …
- Which term of 8, 11, 14, … is 101?
- Find the middle term between 17 and 41 in an arithmetic sequence of three terms.
Answers
1. 147. 2. 46-6n. 3. 32nd term. 4. 29.
Connected routes
Use From Pattern Spotting to a General Term first, then continue to Quadratic Patterns and Recursive Rules and Explicit Rules. Return to the Mathematics Learning Hub.