Question

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x², b², y² are in A.P.

Answer 1

$$\begin{gathered} a,b\mathrm{and}c\mathrm{are}\mathrm{in}\mathrm{A}.\mathrm{P}. \\ \therefore 2b=a+c.......\left(\mathrm{i}\right) \\ a,x\mathrm{and}b\mathrm{are}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore x^2=ab.......\left(\mathrm{ii}\right) \\ \mathrm{And},b,y\mathrm{and}c\mathrm{are}\mathrm{also}\mathrm{in}\mathrm{G}.\mathrm{P}. \\ \therefore y^2=bc.......\left(\mathrm{iii}\right) \\ \mathrm{Now},\mathrm{putting}\mathrm{the}\mathrm{values}\mathrm{of}a\mathrm{and}c: \\ \Rightarrow 2b=\frac{x^2}{b}+\frac{y^2}{b} \\ \Rightarrow 2b^2=x^2+y^2 \\ \mathrm{Therefore},x^2,b^2\mathrm{and}{y}^2\mathrm{are}\mathrm{also}\mathrm{in}A.\mathrm{P}. \end{gathered}$$