Question

The value of xyz is $\frac{15}{2}or\frac{18}{5}$ according as the series a, x, y, z b is arithmetic or harmonic then ab =

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The correct option is C 3

$a,x,y,zinA.P.\Rightarrow x=\frac{3a+b}{4},y=\frac{a+b}{2},z=\frac{a+3b}{4},xyz=\frac{15}{2}\Rightarrow \frac{(3a+b)(a+b)(a+3b)}{32}=\frac{15}{2},a,x,y,zbubH.P.\Rightarrow \frac{1}{a},\frac{1}{x},\frac{1}{y},\frac{1}{b}inA.P.\Rightarrow \frac{1}{x}=\frac{\frac{3}{a}+\frac{1}{b}}{4}=\frac{a+3b}{4ab},\frac{1}{y}=\frac{\frac{1}{a}+\frac{1}{b}}{2}=\frac{a+b}{2ab},\frac{1}{z}=\frac{\frac{1}{a}+\frac{3}{b}}{4}=\frac{3a+b}{4ab}xyz=\frac{32(ab{)}^3}{(a+3b)(a+b)(3a+b)}=\frac{18}{5}Multiplying\left(1\right)and\left(2\right)gives(ab{)}^3=\frac{15}{2}.\frac{18}{5}=27$