An arithmetic sequence is a list of numbers that changes by the same amount each step. That repeated change is called the common difference, and it makes the sequence predictable. Arithmetic sequences appear in stair-step patterns, savings plans, seating arrangements, and many linear models.
Learning them helps connect patterns, formulas, graphs, and real-world change.
Understanding Math: Arithmetic Sequences
A sequence can look regular without being arithmetic. The safest test is to subtract each term from the next term and compare the results. If every result matches, the pattern has a constant rate of change.
A multiplication pattern fails this test because its gaps grow or shrink. For example, the gaps in three, six, twelve, and twenty four are three, six, and twelve. This is a geometric pattern, not an arithmetic one.
Negative differences matter too. A sequence such as fifty, forty five, forty, and thirty five is arithmetic because it falls by five each step. Zero is a valid common difference, so a repeated list such as seven, seven, seven is arithmetic.
There are two useful ways to generate terms. A step by step rule is best when finding nearby terms. Start from a known value, then keep adding the same signed amount.
A direct rule is better when the requested term is far away. It avoids writing every earlier term. The key idea is that the first term needs zero jumps to reach itself.
The second term needs one jump. The tenth term needs nine jumps. This is why students often make an off by one error when using a direct rule.
Count the gaps between terms, not just the term number. Write a few early terms before calculating a distant one to check that the rule starts correctly.
Arithmetic sequences are especially useful when equal changes happen at equal time intervals. A weekly savings plan can begin with a balance and increase by the same deposit each week. Stadium rows may gain the same number of seats in every new row.
A taxi fare with a fixed charge plus the same cost for each kilometre can be modelled in a similar way. The starting amount and the change per interval must have clear units.
A balance is measured in money, while the difference might be money per week. Mixing units can give a sensible looking number with the wrong meaning.
Adding many terms has a useful shortcut because terms at opposite ends form equal pairs. In a list that rises evenly, the first and last terms have the same total as the second and second to last terms. Each pair has one low value and one high value.
This explains why the total can be found from the number of terms and the average of the first and last values. On a graph, use term number on the horizontal axis and term value on the vertical axis. The plotted points lie on a straight line, though the sequence itself is separate points because term numbers are usually whole numbers.
The line rises for a positive difference, falls for a negative difference, and stays flat for zero difference. Check the slope and starting point carefully, since they show the rate of change and initial value.
Key Facts
- Common difference: d = a_n - a_(n-1)
- Explicit formula: a_n = a_1 + (n - 1)d
- Recursive formula: a_n = a_(n-1) + d, with starting value a_1
- Sum of first n terms: S_n = n(a_1 + a_n)/2
- Alternative sum formula: S_n = n/2[2a_1 + (n - 1)d]
- An arithmetic sequence has a linear graph when term number n is plotted against term value a_n.
Vocabulary
- Arithmetic sequence
- A sequence in which each term is found by adding the same constant value to the previous term.
- Common difference
- The constant amount d added to each term to get the next term in an arithmetic sequence.
- Term
- A number in a sequence, usually named by its position such as a_1, a_2, or a_n.
- Explicit formula
- A formula that gives any term of a sequence directly using its term number n.
- Recursive formula
- A formula that defines a term using one or more previous terms and a starting value.
Common Mistakes to Avoid
- Using n instead of n - 1 in the explicit formula. This is wrong because the first term has zero common-difference steps from itself, so a_n = a_1 + (n - 1)d.
- Subtracting terms in the wrong order when finding d. The common difference should be next term minus previous term, such as d = a_2 - a_1.
- Confusing an arithmetic sequence with a geometric sequence. Arithmetic sequences add a constant difference, while geometric sequences multiply by a constant ratio.
- Using the sum formula with the wrong last term. In S_n = n(a_1 + a_n)/2, a_n must be the nth term being included, not just any term from the sequence.
Practice Questions
- 1 The sequence is 7, 11, 15, 19, ... Find the common difference and the 25th term.
- 2 An arithmetic sequence has a_1 = 4 and d = 3. Find S_20, the sum of the first 20 terms.
- 3 A student says the sequence 2, 5, 9, 14, 20 is arithmetic because it is increasing. Explain why this reasoning is incorrect.