Mathematics
Grade 9
15 min
Write variable expressions for geometric sequences
Write variable expressions for geometric sequences
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Introduction & Learning Objectives
Learning Objectives
Define a geometric sequence and identify its key components: the first term (a₁) and the common ratio (r).
Differentiate between a geometric sequence and an arithmetic sequence.
Write the explicit formula for a geometric sequence using variables (a_n = a_1 * r^{n-1}).
Construct a variable expression for the nth term of any given geometric sequence.
Use their derived variable expression to find the value of any term in a sequence.
Model a real-world scenario involving geometric growth or decay with a variable expression.
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In this tutorial, you will learn how to write a single, powerful algebr...
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Key Concepts & Vocabulary
TermDefinitionExample
SequenceAn ordered list of numbers, often following a specific pattern.The list 2, 5, 8, 11, ... is a sequence.
TermEach individual number in a sequence. The position of a term is denoted by a subscript, like a₃ for the third term.In the sequence 4, 8, 16, 32, ..., the third term (a₃) is 16.
Geometric SequenceA sequence where each term is found by multiplying the previous term by a constant, non-zero value.3, 6, 12, 24, ... is a geometric sequence because you multiply by 2 each time.
First Term (a₁)The starting value or the very first number in a sequence.In the sequence 5, 15, 45, ..., the first term (a₁) is 5.
Common Ratio (r)The fixed number you multiply by to get from one term to the next in a geometric sequence.In the sequence 100, 50, 25, ..., the common ratio...
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Core Formulas
Finding the Common Ratio (r)
r = a_n / a_{n-1}
To find the common ratio, divide any term by its immediately preceding term. The result should be constant for the entire sequence.
The Explicit Formula for a Geometric Sequence
a_n = a_1 * r^{n-1}
This is the fundamental formula for writing a variable expression for a geometric sequence. Here, 'a_n' is the nth term, 'a₁' is the first term, 'r' is the common ratio, and 'n' is the term's position number.
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Challenging
A scientist observes a cell culture that starts with 5000 cells. The number of cells grows by 10% every hour. Which expression models the number of cells after the nth hour?
A.a_n = 5500 * (1.1)^(n-1)
B.a_n = 5000 * (1.1)^n
C.a_n = 5000 * (0.1)^n
D.a_n = 5000 * (1.1)^(n-1)
Challenging
For the sequence defined by the variable expression a_n = 5 * 2^(n-1), which term number 'n' will have a value of 640?
A.n = 6
B.n = 7
C.n = 8
D.n = 9
Challenging
Sequence G is defined by a_n = 4 * 2^(n-1). Sequence H is defined by a_n = 256 * (1/2)^(n-1). For what value of n are the terms of the two sequences equal?
A.n = 4
B.n = 5
C.n = 6
D.The terms are never equal.
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