Key takeaway: An arithmetic sequence is a list of numbers that climbs or falls by one constant amount at every step, and that amount is its common difference.
Grade 9 – Algebra 1F-IF.A.3Functions and Graphs
Definition of Arithmetic Sequence
In an arithmetic sequence, the gap between neighbors never changes. The sequence 17, 25, 33, 41 goes up by 8 every time, and 90, 84, 78 goes down by 6 every time. That fixed gap is the common difference \(d\).
Because each step adds the same amount, an arithmetic sequence behaves like a linear function whose inputs are counting numbers. Its graph is a set of separate dots lying on a straight line with slope \(d\). The opening term is named \(a_1\), and a distant term can be reached without writing out everything in between.
Formula and notation
Start at the first term and add \(d\) a total of \(n – 1\) times to reach position \(n\):
\[a_n = a_1 + (n – 1)d\]
To get \(d\), subtract a term from the next one. An arithmetic sequence may also be written step by step as \(a_{n+1} = a_n + d\).
Key facts
- The common difference can be positive, negative or zero.
- Subtracting neighboring terms always gives the same number d.
- Its graph is points on a line, so the rule is a linear function of n.
- Starting at the term number n = 1 means the formula uses n minus 1 jumps.
Arithmetic Sequence example
A theater has 17 seats in row 1, 25 seats in row 2, 33 in row 3, and the pattern continues. How many seats are in row 30?
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Common difference: \(25 – 17 = 8\), and \(33 – 25 = 8\). The first term is \(a_1 = 17\).
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Write the rule: \(a_n = 17 + (n – 1)(8)\).
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Put \(n = 30\): \(a_{30} = 17 + 29(8) = 17 + 232 = 249\).
Row 30 has 249 seats.
Common mistake
Watch out
Using the same pattern, Brent writes \(a_n = 17 + 8n\) and finds \(a_{30} = 17 + 240 = 257\).
Why it is wrong
That rule gives \(a_1 = 25\), not the real first row of 17, because it adds 8 once too often. From row 1 to row 30 there are only 29 jumps, so use \(n – 1\): \(17 + 29(8) = 249\).
Related terms and lessons
- Geometric Sequence
- Explicit Formula
- Recursive Formula
- Arithmetic Series
- Linear Function
- Constant of Proportionality
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