27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given.
b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).

{1, 3, 9, 27, 81, ......}

검증된 단계별 안내

Identify the pattern in the given sequence: {1, 3, 9, 27, 81, ...}. Notice how each term relates to the previous term.

Observe that each term is obtained by multiplying the previous term by 3. For example, 3 = 1 \(\times\) 3, 9 = 3 \(\times\) 3, and so on.

Express this relationship as a recurrence relation: \[a_{n} = 3 \times a_{n-1}\] where \[n \geq 2\].

Specify the initial condition, which is the first term of the sequence: \[a_1 = 1\].

Combine the recurrence relation and the initial condition to fully define the sequence: \[a_1 = 1, \quad a_n = 3a_{n-1} \text{ for } n \geq 2\].

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Sequences and Terms

A sequence is an ordered list of numbers defined by a specific rule. Each number in the sequence is called a term, denoted as aₙ, where n indicates the term's position. Understanding how terms relate to their positions is essential for analyzing and generating sequences.

Recurrence Relation

A recurrence relation expresses each term of a sequence as a function of one or more previous terms. It provides a way to generate the sequence step-by-step, starting from initial term(s). Identifying the recurrence relation helps in understanding the pattern and behavior of the sequence.

Initial Conditions

Initial conditions specify the starting term(s) of a sequence, which are necessary to uniquely determine all subsequent terms using the recurrence relation. Without these values, the sequence cannot be fully generated or analyzed.

27–34. Working with sequences Several terms of a sequence {aₙ}ₙ₌₁∞ are given. b. Find a recurrence relation that generates the sequence (supply the initial value of the index and the first term of the sequence).{1, 3, 9, 27, 81, ......}