Question
If $$a, b$$ and $$c$$ are in A.P., and $$p$$ and $$p'$$ are, respectively, A.M. and G.M. between $$a$$ and $$b$$ while $$q, q'$$ are, respectively, the A,M. and G. M. between $$b$$ and $$c$$ then
A.
$${p^2} + {q^2} = p{'^2} + q{'^2}$$
B.
$$pq = p'q'$$
C.
$${p^2} - {q^2} = p{'^2} - q{'^2}$$
D.
None of these
Answer :
$${p^2} - {q^2} = p{'^2} - q{'^2}$$
Solution :
$$\eqalign{
& 2b = a + c;\,\,a,p,b,q,c\,\,{\text{are in A}}{\text{.P.}} \cr
& {\text{Hence, }}p = \frac{{a + b}}{2}{\text{ and }}q = \frac{{b + c}}{2} \cr
& {\text{Again, }}a,p',b,q'{\text{ and }}c = \,{\text{are in G}}{\text{.P.}} \cr
& {\text{Hence, }}p' = \sqrt {ab} \,\,{\text{and }}q' = \sqrt {bc} \cr
& {p^2} - {q^2} = \frac{{\left( {a - c} \right)\left( {a + c + 2b} \right)}}{4} \cr
& = \frac{{\left( {a - c} \right)\left( {2b + 2b} \right)}}{4}\,\,\,\left[ {\because a + c = 2b} \right] \cr
& = \left( {a - c} \right)b = ab - bc = p{'^2} - q{'^2} \cr} $$