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$\frac{1}{{a + \frac{1}{{b + \frac{1}{c}}}}} = \frac{{16}}{{115}}$

$a,b,c \in \mathbb{N}$

Select correct option(s).

(A) a, b, c are in A.P.
(B) a, b, c are in G.P.
(C) a, b, c are in H.P.
(D) a+b+c = bc

Solution

We have, $a + \frac{1}{{b + \frac{1}{c}}} = \frac{{115}}{{16}}$

$ \Rightarrow a + \frac{c}{{bc + 1}} = \frac{{115}}{{15 + 1}}$

$ \Rightarrow a + \frac{c}{{bc + 1}} = \frac{{115}}{{5 \times 3 + 1}}$

$ \Rightarrow a + \frac{c}{{bc + 1}} = \frac{{16 \times 7 + 3}}{{5 \times 3 + 1}}$

$ \Rightarrow a + \frac{c}{{bc + 1}} = 7 + \frac{3}{{5 \times 3 + 1}}$

$\therefore \{ a,b,c\}  \equiv \{ 7,5,3\} $

Hence, (A) & (D).