Question

Write each of the following sets in set-builder notation. $$ \\{2,4,8,16,32,64 \ldots\\} $$

Short answer

The set \{2, 4, 8, 16, 32, 64...\} can be represented in set-builder notation as \{x : x = 2^n, n ∈ N\}

Step-by-step solution

Identify the Pattern

The first step is to identify the pattern followed by the elements in the set. Here, starting from the first number, each subsequent number is obtained by doubling (multiplying by 2) the previous number. This is a geometric sequence with a common ratio of 2.

Write the Set in Set-Builder Notation

After identifying the pattern, the set can now be written in set-builder notation. Here the set is the set of all numbers in the form \(2^n\), where 'n' is a natural number. Therefore, the set can be represented in set-builder notation as \{x : x = 2^n, n ∈ N\}

Key concepts

Geometric Sequence

A geometric sequence is a series of numbers in which each term after the first is found by multiplying the previous term by a constant called the common ratio. In our example, the sequence starts with 2, and each subsequent term is 2 times the term before it, making the common ratio 2. This means that if you pick any number in the sequence and divide it by its predecessor, you'll always get 2.

Geometric sequences can be found in various places in nature and human-made systems, from the spirals of seashells and the branches of trees to the interest calculation in your savings account. It's a fundamental concept in mathematics because it helps us describe and understand these types of growth patterns efficiently and effectively.

• Starting Term: 2
• Common Ratio: 2
• Sequence Formula: The n-th term is given by the formula \(a_n = a_1 \times r^{(n-1)}\), where \(a_1\) is the first term and 'r' is the common ratio.

In our set, the terms can be denoted as \(2, 2^2, 2^3, 2^4, \ldots\), with each term representing \(2^n\), where 'n' is the position of the term in the sequence.

Mathematical Patterns

Identifying mathematical patterns allows us to predict and understand the world around us. Patterns can occur in sequences of numbers, shapes, events, or even abstract ideas. Once we recognize a pattern, we can describe it using a mathematical formula, enabling us to find any term in the sequence without listing all the terms.

Patterns can be arithmetic (where the difference between consecutive terms is constant), geometric (where the quotient of consecutive terms is constant), or more complex. Recognizing the pattern in a sequence is a critical skill, as it provides a foundation for understanding functions, series, and much more in mathematics.

• Arithmetic Sequence: Addition or subtraction by a fixed number
• Geometric Sequence: Multiplication or division by a fixed number
• Complex Patterns: Could involve alternating sequences or varying operations

In our exercise, the sequence exhibits a repeating pattern of multiplication, specifically by the factor of 2. This discovery is what leads us to classify the sequence as a geometric one.

Natural Numbers

Natural numbers are the set of all positive integers starting from 1. They are the numbers we typically use for counting and ordering and are a foundational block in mathematics. When we talk about 'n' being a natural number in our set-builder notation, it restricts the values of 'n' to these counting numbers, excluding zero and negative numbers.

In set theory and other branches of mathematics, natural numbers are often denoted by the symbol \(\mathbb{N}\), and they are infinite. Here are a few characteristics of natural numbers:

• They begin from 1 and go on indefinitely: 1, 2, 3, 4, \ldots
• Used for counting (there are 3 apples) and ordering (this is the 1st item)
• They do not include zero, fractional, or negative numbers

So, when the solution to our exercise specifies 'n ∈ N', it means that 'n' can be any natural number, giving us an infinite set of terms in the geometric sequence.