Question

If $a,a_1,a_2,a_3,....a_{2n-1},b$ are in AP, $a,b_1,b_2,b_3,...,b_{2n-1},b$ are in GP and $a,c_1,c_2,c_3,....,c_{2n-1},b$ are in HP, where a, b are positive, then the equation $a_n{x}^2-b_{n}x+c_n=0$ has its roots

• A. Real and unequal.
• B. Real and equal.
• C. Imaginary
• D. None of these.

Answer 1

Answer: (C)

Imaginary

As odd number of AM, G .M and H.M. are inserted between a & b.
So, middle term of AP is $AM=a_n$
middle term of GP is $GM=b_n$
middle term of HP is $HM=c_n$
$\therefore a_n,b_n,c_n$ are in G.P.
$\therefore D=$ discriminant of quadratic equation < 0
$\therefore$ roots are imaginary.