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.
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.