Question
Let the positive numbers $$a, b, c, d$$ be in A.P. Then $$abc, abd, acd, bcd$$ are
A.
Not in A.P./G.P./H.P.
B.
in A.P.
C.
in G.P.
D.
in H.P.
Answer :
in H.P.
Solution :
$$\eqalign{
& a,b,c,d\,\,{\text{are in A}}{\text{.P}}{\text{.}} \cr
& \therefore \,\,\,\,\,d,c,b,a{\text{ are also in A}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,\,\frac{d}{{abcd}},\frac{c}{{abcd}},\frac{b}{{abcd}},\frac{a}{{abcd}}{\text{ are also in A}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,\,\frac{1}{{abc}},\frac{1}{{abd}},\frac{1}{{acd}},\frac{1}{{bcd}}{\text{ are in A}}{\text{.P}}{\text{.}} \cr
& \Rightarrow \,\,abc,abd,acd,bcd{\text{ are in H}}{\text{.P}}{\text{.}} \cr} $$