Mathematics
Grade 9
15 min
Identify arithmetic and geometric sequences
Identify arithmetic and geometric sequences
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a sequence, a term, an arithmetic sequence, and a geometric sequence.
Identify the common difference in an arithmetic sequence.
Identify the common ratio in a geometric sequence.
Differentiate between arithmetic, geometric, and other types of sequences.
Analyze a given sequence to determine if it is arithmetic by testing for a common difference.
Analyze a given sequence to determine if it is geometric by testing for a common ratio.
Justify the classification of a sequence by showing their calculations.
Have you ever noticed how a savings account grows over time or how a bouncing ball gets lower with each bounce? 📉 These are real-world examples of mathematical patterns called sequences!
In this tutorial, you will learn to identify two of the mos...
2
Key Concepts & Vocabulary
TermDefinitionExample
SequenceAn ordered list of numbers, called terms, that follow a specific pattern or rule.3, 6, 9, 12, ...
TermEach individual number in a sequence. We use notation like a_1 for the first term, a_2 for the second term, and so on.In the sequence 3, 6, 9, 12, ..., the third term (a_3) is 9.
Arithmetic SequenceA sequence in which the difference between any two consecutive terms is constant.2, 7, 12, 17, 22, ... (you add 5 each time)
Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic sequence.In the sequence 2, 7, 12, 17, ..., the common difference is 5.
Geometric SequenceA sequence in which the ratio between any two consecutive terms is constant.3, 6, 12, 24, 48, ... (you multiply by 2 each time)
Common Ratio (r)The c...
3
Core Formulas
Test for an Arithmetic Sequence
d = a_{n} - a_{n-1}
To determine if a sequence is arithmetic, subtract any term from the term that follows it. If this difference is the same for all consecutive pairs of terms, the sequence is arithmetic.
Test for a Geometric Sequence
r = a_{n} / a_{n-1}
To determine if a sequence is geometric, divide any term by the term that precedes it. If this ratio is the same for all consecutive pairs of terms, the sequence is geometric.
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Challenging
Consider a sequence defined by the rule a_n = n^2 + 1, where n is the term number (1, 2, 3, ...). The first few terms are 2, 5, 10, 17, ... How would you classify this sequence?
A.Arithmetic
B.Geometric
C.Both arithmetic and geometric
D.Neither arithmetic nor geometric
Challenging
For what value of x will the terms x-1, 2x, and 5x-8 form the first three terms of an arithmetic sequence?
A.2
B.4
C.5
D.7
Challenging
For what positive value of k will the terms k-2, k, and 2k-1 form a geometric sequence?
A.1
B.4
C.3
D.2
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing Free