Let (an) be a sequence defined bya1 = 12, an+1 = 2 − 2an + 1 for n ≥ 1.(a) Show that an ∈ [0, 1] for all n. [4 marks](b) Show that (an) is increasing. [5 marks](c) Show that (an) is convergent, and calculate its limit.
Solution
(a) We will prove by induction that an ∈ [0, 1] for all n.
Base case: a1 = 12 is not in the interval [0, 1]. Therefore, the statement is false.
(b) To show that (an) is increasing, we need to show that an+1 > an for all n.
Given the recursive formula an+1 = 2 - 2an + 1, we can rewrite this as an+1 - an = 2 - 3an.
For the sequence to be increasing, we need 2 - 3an > 0, which simplifies to an < 2/3.
However, from part (a), we know that an is not necessarily in the interval [0, 1] for all n. Therefore, we cannot definitively say that the sequence is increasing.
(c) A sequence is convergent if it approaches a certain value.
Given the recursive formula an+1 = 2 - 2an + 1, if the sequence is convergent to a limit L, then we should have L = 2 - 2L + 1.
Solving this equation gives L = 1/3.
However, since we have not established that the sequence is increasing and bounded, we cannot definitively say that the sequence is convergent.
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