Identify each set as finite or infinite. Then determine whether 10 is an element of the set. {1, 1/2, 1/4, 1/8, ....}

Verified step by step guidance

1. First, recognize the pattern of the set: {1, 1/2, 1/4, 1/8, ...} is a sequence where each term is half of the previous term.
2. Since the pattern continues indefinitely by repeatedly dividing by 2, the set contains infinitely many elements, so it is an infinite set.
3. Next, to determine if 10 is an element of the set, consider the general term of the sequence. The nth term can be written as \(a_n = \frac{1}{2^{n-1}}\).
4. Check if there exists an integer \(n\) such that \(a_n = 10\), which means solving \(\frac{1}{2^{n-1}} = 10\).
5. Since \(\frac{1}{2^{n-1}}\) is always less than or equal to 1 for all positive integers \(n\), it can never equal 10, so 10 is not an element of the set.

Key Concepts

Finite vs. Infinite Sets

A finite set contains a limited number of elements, while an infinite set has no end and continues indefinitely. Recognizing whether a set is finite or infinite helps in understanding its size and behavior.

Set Notation and Elements

Set notation uses curly braces to list elements or describe a pattern. Understanding how to interpret the elements and the notation is essential to determine membership and properties of the set.

Membership in a Set

Determining if an element belongs to a set involves checking if the element fits the pattern or is explicitly listed. This concept is key to answering whether a specific number, like 10, is part of the given set.