If a1=2 and a_{n+1} = (a_n + 6)/2, what is the limit of a_n as n → ∞?

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Prepare for the AMSOC 26-003 Module A Test. Utilize flashcards and multiple choice questions with hints and explanations. Ace your exam!

Multiple Choice

If a1=2 and a_{n+1} = (a_n + 6)/2, what is the limit of a_n as n → ∞?

Explanation:
This sequence moves halfway toward 6 each step, so if a limit L exists it must satisfy the fixed-point condition L = (L + 6)/2. Solving 2L = L + 6 gives L = 6. From the start a1 = 2, each new term is the average with 6, so a_{n+1} = (a_n + 6)/2 is greater than a_n while a_n remains below 6. It also keeps a_{n+1} below 6 because (a_n + 6)/2 < 6 when a_n < 6. Thus the sequence is increasing and bounded above by 6, so it converges to the fixed point 6.

This sequence moves halfway toward 6 each step, so if a limit L exists it must satisfy the fixed-point condition L = (L + 6)/2. Solving 2L = L + 6 gives L = 6. From the start a1 = 2, each new term is the average with 6, so a_{n+1} = (a_n + 6)/2 is greater than a_n while a_n remains below 6. It also keeps a_{n+1} below 6 because (a_n + 6)/2 < 6 when a_n < 6. Thus the sequence is increasing and bounded above by 6, so it converges to the fixed point 6.

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