Question

Suppose a, b, c are in

a.p. and a2 , b2 , c2 are in g.p. if a < b < c and a + b + c = 3/2, then the value of a is

Answer 1

Since a + c = 2b, b = 1/2. Also b2b2 = ± ac ⇒ ac = ± 1/4 and a + c = 1
Hence a, c are the roots of the equation x2x2 – x ± 1/4 = 0

⇒a=1/2−1/√2

Answer 2

Answer: 1/2 - 1/√2

Explanation: a, b, c are in AP .

Therefore a = b-d and c= b+d

Given, a+b+c = 3/2

           b= 1/2

Hence, a=1/2-d, b=1/2,c=1/2+d

Given, a²,b²,c² are in GP.

(b²)²= a²c²

Solving the above equation, we get,

(1/2-d)(1/2+d)= (±1/4)

Solve for d, and we get d= 1/√2 (d≥0)

a = 1/2 - 1/√2