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

검증된 단계별 안내

Observe the given set: \(\{ 1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \ldots \}\). Notice the pattern in the terms.

Recognize that each term after the first is obtained by multiplying the previous term by \(\frac{1}{3}\). This means the set is a geometric sequence with the first term \(a = 1\) and common ratio \(r = \frac{1}{3}\).

Understand that the ellipsis (\(\ldots\)) at the end of the set indicates the sequence continues indefinitely, without stopping at a last term.

Recall the definitions: a finite set has a limited number of elements, while an infinite set continues without end.

Conclude that since the sequence continues indefinitely, the set is infinite.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Sequences

A sequence is an ordered list of numbers following a specific pattern. Each number in the sequence is called a term, and sequences can be finite (having a limited number of terms) or infinite (continuing indefinitely). Understanding the nature of the sequence helps determine its classification.

Geometric Sequence

A geometric sequence is a sequence where each term after the first is found by multiplying the previous term by a constant ratio. For example, in the sequence {1, 1/3, 1/9, 1/27, ...}, each term is multiplied by 1/3. Recognizing this pattern is key to analyzing the sequence.

Finite vs. Infinite Sets

A finite set contains a limited number of elements, while an infinite set has no end and continues indefinitely. Determining whether a sequence is finite or infinite depends on whether the pattern stops or goes on forever, which is essential for classifying the given set.