Question

If A₁, A₂ be two AM's and G₁, G₂ be two GM's between a and b, then find the value of \[\frac{A_1 + A_2}{G_1 G_2}\]

Answer 1

\[\text {It is given that A_1 and A_2 are the A . M . s between a and b } . \]
\[\text{ Thus, a , A_1 , A_2 and b are in A . P . with common difference d  }. \]
\[\text{ Here }, d = \frac{b - a}{3}\]
\[ \therefore A_1 = a + \frac{b - a}{3} = \frac{2a + b}{3}\]
\[\text{ and } A_2 = a + \frac{2\left( b - a \right)}{3} = \frac{a + 2b}{3}\]
\[\text{ It is also given that G_1 and G_2 are the G . M . s between a and b } . \]
\[\text{ Thus, a , G_1 , G_2 and b are in G . P . with common ratio r } . \]
\[\text{ Here }, r = \left( \frac{b}{a} \right)^\frac{1}{3} \]
\[ \therefore G_1 = a \left( \frac{b}{a} \right)^\frac{1}{3} = b^\frac{1}{3} a^\frac{1}{3} \]
\[\text{ and } G_2 = a \left[ \left( \frac{b}{a} \right)^\frac{1}{3} \right]^2 = b^\frac{1}{3} a^\frac{1}{3} \]
\[ \Rightarrow \frac{A_1 + A_2}{G_1 G_2} = \frac{\frac{2a + b}{3} + \frac{a + 2b}{3}}{b^\frac{1}{3} a^\frac{1}{3} \times b^\frac{1}{3} a^\frac{1}{3}} = \frac{a + b}{ab}\]
\[\]