Check whether or not the sequence , defined by:and

Question

Check whether or not the sequence $(a_n)_{n \in \mathbb{N}}$ , defined by:
$$a_{n+1} = 2 - \frac{1}{a_n + 2} \quad \forall n \ge 1$
and $a_1 = 2$ is convergent.

20 Jan 2026

Answer :

Word Count : 158

The sequence is defined by (a_1=2) and (a_{n+1}=2-\dfrac{1}{a_n+2}) for all (n\ge1). First compute a few terms to observe its behavior. Since (a_1=2), we have [ a_2=2-\frac{1}{2+2}=2-\frac14=\frac74, __________ _______ _________ __________ _____.
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