Use set notation, and list all the elements of each set. {1, 1/2, 1/4, ...., 1/32} .

Verified step by step guidance

1. Identify the pattern in the given set: {1, 1/2, 1/4, ..., 1/32}.
2. Recognize that each element is a power of 1/2: 1 = (1/2)^0, 1/2 = (1/2)^1, 1/4 = (1/2)^2, ..., 1/32 = (1/2)^5.
3. List the elements of the set using the identified pattern: {1, 1/2, 1/4, 1/8, 1/16, 1/32}.
4. Express the set in set notation: { (1/2)^n | n is an integer, 0 ≤ n ≤ 5 }.
5. Verify that all elements from the original set are included in the listed set.

Key Concepts

Set Notation

Set notation is a mathematical way to describe a collection of distinct objects, known as elements. It typically uses curly braces to enclose the elements, which can be numbers, symbols, or other sets. Understanding set notation is essential for identifying and listing elements accurately, as it provides a clear framework for organizing and communicating mathematical ideas.

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the given set, the elements are generated by repeatedly multiplying by 1/2, starting from 1. Recognizing this pattern is crucial for determining the elements of the set and understanding its structure.

Listing Elements

Listing elements involves explicitly writing out all the members of a set. For finite sets, this means identifying each unique element without repetition. In the context of the given question, it requires recognizing the pattern of the geometric sequence and calculating each term until the specified limit, ensuring that all elements are included in the final representation.