Geometric Sequence: Understanding 1/8, 1/4, 1/2
In mathematics, a geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant. This constant is called the common ratio (r). In this article, we will explore the concept of geometric sequences with a focus on the sequence 1/8, 1/4, 1/2.
What is a Geometric Sequence?
A geometric sequence is a sequence of numbers where each term is obtained by multiplying the previous term by a fixed constant (r). The general formula for a geometric sequence is:
an = ar^(n-1)
where:
- an is the nth term of the sequence
- a is the first term (also called the initial term)
- r is the common ratio
- n is the term number (starts at 1)
The Sequence 1/8, 1/4, 1/2
Let's take a closer look at the sequence 1/8, 1/4, 1/2. Can we identify the common ratio (r) and the initial term (a)?
Identifying the Common Ratio (r)
To find the common ratio (r), we can divide each term by its previous term:
- 1/4 ÷ 1/8 = 2
- 1/2 ÷ 1/4 = 2
The common ratio (r) is 2.
Identifying the Initial Term (a)
The initial term (a) is the first term of the sequence, which is 1/8.
General Formula
Now that we have the common ratio (r) and the initial term (a), we can write the general formula for this geometric sequence:
an = (1/8) × 2^(n-1)
Properties of Geometric Sequences
Geometric sequences have several important properties:
- The terms of a geometric sequence either increase or decrease by a constant factor.
- The sum of a geometric sequence can be calculated using the formula:
- S = (a × (1 - r^n)) / (1 - r)
- where S is the sum of the sequence, a is the initial term, r is the common ratio, and n is the number of terms.
Real-World Applications
Geometric sequences have many real-world applications, such as:
- Biology: Modeling population growth or decline
- Finance: Calculating compound interest
- Computer Science: Algorithm design and analysis
Conclusion
In this article, we explored the concept of geometric sequences with a focus on the sequence 1/8, 1/4, 1/2. We identified the common ratio (r) and the initial term (a), and wrote the general formula for this geometric sequence. We also discussed the properties of geometric sequences and their real-world applications.
