Mathematics
Grade 11
15 min
Write a formula for a geometric sequence
Write a formula for a geometric sequence
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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).
Calculate the common ratio from a given geometric sequence.
Write the explicit formula for a geometric sequence given the first term and the common ratio.
Derive the explicit formula for a geometric sequence when given any two terms.
Use the explicit formula to find the value of any term (aₙ) in a sequence.
Distinguish between a geometric sequence and an arithmetic sequence.
Ever wonder how a bank calculates compound interest or how a bouncing ball loses height with each bounce? 💰 These are real-world examples of geometric sequences!
This tutorial will teach you how to capture the pattern of repeated multiplication found in geometric sequences usin...
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Key Concepts & Vocabulary
TermDefinitionExample
SequenceAn ordered list of numbers, where each number is called a term.The list 5, 10, 15, 20, ... is a sequence.
Geometric SequenceA sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number.In the sequence 2, 6, 18, 54, ..., each term is multiplied by 3 to get the next term.
First Term (a₁)The starting value or the very first number in a sequence.In the sequence 100, 50, 25, ..., the first term, a₁, is 100.
Common Ratio (r)The constant factor you multiply by to get from one term to the next. It can be found by dividing any term by its preceding term (r = aₙ / aₙ₋₁).In the sequence 3, -6, 12, -24, ..., the common ratio, r, is -2 because -6 / 3 = -2.
nth Term (aₙ)A variable that represents the term in the nth posit...
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Core Formulas
The Explicit Formula for a Geometric Sequence
aₙ = a₁ * r^(n-1)
Use this formula to find the value of any term (aₙ) in a geometric sequence. You need the first term (a₁), the common ratio (r), and the position of the term you want to find (n).
Finding the Common Ratio
r = aₙ / aₙ₋₁
To find the common ratio (r), divide any term in the sequence by the term that comes directly before it. This is often the first step in writing the formula for a sequence.
4 more steps in this tutorial
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Challenging
The 3rd term of a geometric sequence is 20 and the 5th term is 80. Which of the following is a possible explicit formula for the sequence?
A.aₙ = 5 * (-2)^(n-1)
B.aₙ = 20 * (2)^(n-1)
C.aₙ = 5 * (4)^(n-1)
D.aₙ = 10 * (-2)^(n-1)
Challenging
The 4th term of a geometric sequence is -54 and the 7th term is 1458. What is the explicit formula for the sequence?
A.aₙ = -2 * (3)^(n-1)
B.aₙ = 2 * (-3)^(n-1)
C.aₙ = 3 * (-2)^(n-1)
D.aₙ = -3 * (2)^(n-1)
Challenging
In a geometric sequence, a₂ = 2 and a₆ = 1/8. If the common ratio is positive, what is the explicit formula?
A.aₙ = 2 * (1/2)^(n-1)
B.aₙ = 1 * (1/4)^(n-1)
C.aₙ = 4 * (1/2)^(n-1)
D.aₙ = 8 * (1/4)^(n-1)
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