Arithmetic & Geometric Sequences
Sequences are ordered lists of numbers following a rule; arithmetic sequences have a constant difference between terms while geometric sequences have a constant ratio, both with real-world applications.
What You Need to Know
Key Concept Diagram
An arithmetic sequence has a common difference d: each term is found by adding d to the previous term. The nth term is T(n) = a + (n−1)d
A geometric sequence has a common ratio r: each term is found by multiplying the previous term by r. The nth term is T(n) = ar^(n−1)
The sum of an arithmetic sequence: S(n) = n/2 × (2a + (n−1)d) or S(n) = n/2 × (first + last)
Geometric sequences model exponential growth (r > 1) or decay (0 < r < 1) in real contexts such as population and depreciation
Key Vocabulary
Arithmetic sequence
A sequence where consecutive terms differ by a fixed constant called the common difference
Geometric sequence
A sequence where each term is obtained by multiplying the previous term by a fixed constant called the common ratio
Common difference (d)
The constant amount added between consecutive terms in an arithmetic sequence
Common ratio (r)
The constant factor multiplied between consecutive terms in a geometric sequence
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the 10th term of the arithmetic sequence 3, 7, 11, 15, …?
Question 2
The sequence 2, 6, 18, 54, … is geometric. What is the 5th term?
Question 3
Find the sum of the first 8 terms of the arithmetic sequence 5, 9, 13, 17, …
Key Concepts Summary
- ●An arithmetic sequence has a common difference d: each term is found by adding d to the previous term. The nth term is T(n) = a + (n−1)d
- ●A geometric sequence has a common ratio r: each term is found by multiplying the previous term by r. The nth term is T(n) = ar^(n−1)
- ●The sum of an arithmetic sequence: S(n) = n/2 × (2a + (n−1)d) or S(n) = n/2 × (first + last)
- ●Geometric sequences model exponential growth (r > 1) or decay (0 < r < 1) in real contexts such as population and depreciation