Concept:Arithmetic Progression:If a, b and c are in AP thenb - a = c - b or 2b = a + cGeometric Progression:If a, b and c are in GP thenb/a = c/b or b2 = acHarmonic Progression:If a, b and c are in HP then 1/a, 1/b, and 1/c will be on GPCalculation:Given:\(\displaystyle \frac{a+b}{2}, b, \frac{b+c}{2}\) are in HP? \(\displaystyle \frac{2}{a+b}, \frac{1}{b}, \frac{2}{b+c}\) are AP? \(\displaystyle \frac{1}{b}- \frac{2}{a+b}=\frac{2}{b+c}-????\frac{1}{b}\)? \(\displaystyle \frac{1}{b}+\frac{1}{b}=\frac{2}{b+c}\ +???\frac{2}{a+b}\)? \(\displaystyle \frac{2}{b}=2\left[\frac{1}{b+c}\ +???\frac{1}{a+b}\right] \)? \(\displaystyle \frac{2}{b}=2\left[\frac{a+b+b+c}{(a+b)(b+c)}\right] \)? \(\displaystyle \frac{1}{b}=\left[\frac{a+2b+c}{(a+b)(b+c)}\right] \)? (a + b)(b + c) = ab + 2b2 + bc? ab + ac + bc + b2 = ab + 2b2 + bc? b2 = acSince b2 = ac, b is the geometric mean of a and c.? a, b, c are in GP