Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 5,aₙ₊₁ = √(5aₙ)

검증된 단계별 안내

Identify the recursive sequence given: \(a_1 = 5\) and \(a_{n+1} = \sqrt{5a_n}\). We want to find the limit \(L\) as \(n\) approaches infinity, assuming the sequence converges.

Assuming the sequence converges to a limit \(L\), then both \(a_n\) and \(a_{n+1}\) approach \(L\). So, set \(L = \sqrt{5L}\) to find the limit.

Square both sides of the equation to eliminate the square root: \(L^2 = 5L\).

Rearrange the equation to standard polynomial form: \(L^2 - 5L = 0\).

Factor the equation: \(L(L - 5) = 0\). The possible limits are \(L = 0\) or \(L = 5\). Use the initial term and the nature of the sequence to determine which limit is valid.

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.

Recursively Defined Sequences

A recursively defined sequence is one where each term is defined based on one or more previous terms. Understanding how to work with such sequences involves using the given formula repeatedly to generate terms and analyze their behavior as n increases.

Convergence of Sequences

A sequence converges if its terms approach a specific finite value as n approaches infinity. Determining convergence often involves finding a limit L such that the sequence terms get arbitrarily close to L, which is essential for solving limit problems in recursive sequences.

Finding Limits of Recursive Sequences

To find the limit of a convergent recursive sequence, assume the limit exists and set L equal to the limit of both aₙ and aₙ₊₁. Then solve the resulting equation, often involving algebraic manipulation, to find the value of L that satisfies the recursive definition.

Recursively Defined SequencesIn Exercises 101–108, assume that each sequence converges and find its limit.a₁ = 5,aₙ₊₁ = √(5aₙ)