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# Let a, b, be positive real numbers. If a, A1, A2 b are in arithmetic progression, a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression,show that G1 G=G2/H1 H2 = A1 + A2/ H1 H2 = (2a + b) (a + 2b)/ 9ab

6 years ago
Hello Student,
Clearly A1 + A2 = a + b
1/H1 + 1/H2 = 1/a + 1/b
⇒ H1 + H2/H1 H2 = a + b/ab = A1 + A2/G1 G2
⇒ G1 G2 / H1 H2 = A1 + A2/H1 + H2
Also 1/H1 = 1/a + 1/3 (1/b – 1/a) ⇒ H1 = 3ab/2b + a
1/H2 = 1/a + 2/3(1/b – 1/a) ⇒ H2 = 3ab/2a + b
⇒ A1 + A2/H1 + H2 = $\frac{a+b}{3ab\left ( \frac{1}{2b+a}+\frac{1}{2a+b} \right )}$
= $\frac{\left ( 2b+a \right )\left ( 2a+b \right )}{9ab}$

Thanks