Identify the set { 1,1/3, 1/9 ,1/27, ....} as finite or infinite.

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. To determine whether a set is finite or infinite, one must analyze the pattern of its elements. If the elements can be counted and there is a last element, the set is finite; otherwise, it is infinite.

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, each term is obtained by multiplying the previous term by 1/3, indicating that it follows a geometric pattern. Understanding this helps in identifying the nature of the set.

Limit of a Sequence

The limit of a sequence refers to the value that the terms of the sequence approach as the index (or term number) goes to infinity. In the context of the given set, as the terms progress, they approach zero but never actually reach it. This characteristic of the sequence contributes to its classification as infinite, as it continues indefinitely without terminating.