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 and its terms.
Define an arithmetic sequence and calculate its common difference.
Define a geometric sequence and calculate its common ratio.
Differentiate between arithmetic and geometric sequences by analyzing the pattern between consecutive terms.
Classify a given sequence as arithmetic, geometric, or neither.
Apply the correct test (subtraction or division) to identify a sequence type.
Ever notice how a stadium's seats are arranged in rows with more seats in each successive row? That's a sequence in action! 🏟️
In this lesson, we will explore two of the most important types of mathematical patterns: arithmetic and geometric sequences. You will learn how to spot these patterns, identify their key features, and distinguish th...
2
Key Concepts & Vocabulary
TermDefinitionExample
SequenceAn ordered list of numbers. Each number in the list is called a term.3, 7, 11, 15, ... is a sequence. The first term is 3.
Arithmetic SequenceA sequence in which the difference between any two consecutive terms is constant. This is like a linear pattern.2, 8, 14, 20, ... (you add 6 each time)
Common Difference (d)The constant value that is added to each term to get the next term in an arithmetic sequence. It can be positive or negative.In the sequence 2, 8, 14, 20, ..., the common difference is 6. In 50, 40, 30, ..., the common difference is -10.
Geometric SequenceA sequence in which the ratio between any two consecutive terms is constant. This is like an exponential pattern.3, 9, 27, 81, ... (you multiply by 3 each time)
Common Ratio (r)The constant value th...
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Core Formulas
Test for an Arithmetic Sequence
a_n - a_{n-1} = d
To check if a sequence is arithmetic, subtract any term from the term that comes right after it. If the result is always the same constant value (d), the sequence is arithmetic.
Test for a Geometric Sequence
a_n / a_{n-1} = r
To check if a sequence is geometric, divide any term by the term that comes right before it. If the result is always the same constant value (r), the sequence is geometric.
5 more steps in this tutorial
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Challenging
If a sequence is defined such that a_n - a_{n-1} = a_{n-1} - a_{n-2} for all terms n > 2, what must be true about the sequence?
A.The sequence is arithmetic.
B.The sequence is geometric.
C.The sequence is neither arithmetic nor geometric.
D.All terms in the sequence are equal.
Challenging
A sequence is generated by the rule a_n = 5n - 3 for n = 1, 2, 3, ... What type of sequence is this?
A.Geometric
B.Arithmetic
C.Neither
D.Cannot be determined
Challenging
A sequence is generated by the rule a_n = 4 * (2)^(n-1) for n = 1, 2, 3, ... What type of sequence is this?
A.Arithmetic
B.Neither
C.Geometric
D.Cannot be determined
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