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. Step 1: Identify the pattern in the set.
2. Step 2: Recognize that the set is a geometric sequence with a common ratio.
3. Step 3: Determine if the set is finite or infinite by analyzing the pattern.
4. Step 4: Check if 10 is an element of the set by comparing it to the terms in the sequence.
5. Step 5: Conclude whether the set is finite or infinite and if 10 is 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 bounds and continues indefinitely. For example, the set of natural numbers is infinite, as it goes on forever. In contrast, the set {1, 2, 3} is finite because it contains only three elements.

Elements of a Set

An element of a set is an individual object or number that belongs to that set. For instance, in the set {1, 2, 3}, the number 2 is an element. To determine if a number is an element of a set, one must check if it appears within the defined collection of items.

Understanding Infinite Sequences

An infinite sequence is a list of numbers that continues indefinitely, often defined by a specific rule or pattern. The set {1, 1/2, 1/4, 1/8, ...} represents a geometric sequence where each term is half of the previous one. Recognizing the pattern helps in identifying the nature of the sequence and its elements.