IGNOU BMTC 133 2026 - Check whether or not the sequence , defined by:and &nbsp

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, ] [ a_3=2-\frac{1}{\frac74+2}=2-\frac{1}{\frac{15}{4}}=2-\frac{4}{15}=\frac{26}{15}. ] These values are all less than 2 and positive. Assume (a_n>0). Then (a_n+2>2),

___ _____ ____ _________ _________ ___ _____.
________ _______ ________ ________ __________ _________ ________.
__________ ______ ________ __________ ____.
___ _________ _____ ________ _______.
_____ _________ ___ ______ ________ __________ _____ _____ _______ ______.
_________ ______ _______ ______ ___ ____ __________ ________ __________ ________ ___ __________.
___ _____ ______ ______ ______ _________ ____ __________ ________ ______ ___.
______ ______ ________ _______ ______ _____ ___ ________ ____ ______ ____ ________.
____ __________ __________ __________ ____ __________ _________ _____.
_______ ____ __________ __________ ____ _______ _____ _______ _______ __________ __________.
_____ _______ _____ __________ ______ ______ _________ ________.
____ ___ _____ _____ _____ __________.
____ ________ ________ _______ _____ _______ _____ __________ __________ ________ ____.
___ _____.

IGNOU ACADEMY

Your IGNOU journey, made easier.