If H1,H2 are two harmonic means between two positive numbers a and b(a≠ b),A and G are the arithmetic and geometric means between a and b, then (H2+H1/H2H1) is

Question

If H1,H2 are two harmonic means between two positive numbers a and b(a≠ b),A and G are the arithmetic and geometric means between a and b, then (H2+H1/H2H1) is (A)  (A/G)  (B)  (2A/G)  (C)  (A/2G2)  (D)  (A/G2)

Tardigrade Admin · Accepted Answer

Since, H1 and H2 are two harmonic means between two positive numbers a and b, then a,H1,H2,b are in HP. ∴ H1=(3ab/a+2b),H2=(3ab/2a+b) Now, (H1+H2/H1H2)=(1/H2)+(1/H1) =(2a+b/3ab)+(a+2b/3ab) =(3a+3b/3ab)=(a+b/ab) .. (i) Now, A is the arithmetic mean between a and b, then 2A=a+b ...(ii) and G is the geometric mean between a and b, then ab=G2 ...(iii) From Eqs. (i), (ii) and (iii), we get (H1+H2/H1H2)=(2A/G2)

Q. If $H_{1}, H_{2}$ are two harmonic means between two positive numbers $a$ and $b (a \neq = b), A$ and $G$ are the arithmetic and geometric means between $a$ and $b$ , then $\frac{H _{2} + H _{1}}{H _{2} H _{1}}$ is

$\frac{A}{G}$

$\frac{2 A}{G}$

$\frac{A}{2 G ^{2}}$

$\frac{A}{G ^{2}}$

$\frac{2 A}{G ^{2}}$

Q. If H1​,H2​ are two harmonic means between two positive numbers a and b(a=b),A and G are the arithmetic and geometric means between a and b, then H2​H1​H2​+H1​​ is

GA​

G2A​

2G2A​