Recursively Defined Sequences
In Exercises 101–108, assume that each sequence converges and find its limit.
a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)

검증된 단계별 안내

Identify the recursive sequence given: \(a_1 = 2\) and \(a_{n+1} = \frac{72}{1 + a_n}\). 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 \(L = \frac{72}{1 + L}\).

Multiply both sides of the equation by \((1 + L)\) to clear the denominator: \(L(1 + L) = 72\).

Rewrite the equation as a quadratic: \(L^2 + L - 72 = 0\).

Solve the quadratic equation for \(L\) using the quadratic formula \(L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=1\), and \(c=-72\). Then determine which root makes sense in the context of the sequence.

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 in terms of one or more previous terms. Understanding how to use the given initial term and the recursive formula is essential to generate terms and analyze the sequence's behavior.

Convergence of Sequences

A sequence converges if its terms approach a specific finite value as the index goes to infinity. Determining convergence involves analyzing the recursive formula and checking if the sequence settles to a stable limit.

Finding the Limit of a Recursive Sequence

To find the limit of a convergent recursive sequence, assume the limit exists and set it equal to the expression defining the next term. Solving the resulting equation yields the limit, which represents the value the sequence approaches.

Recursively Defined SequencesIn Exercises 101–108, assume that each sequence converges and find its limit.a₁ = 2,aₙ₊₁ = 72 / (1 + aₙ)