Correct Answer - Option 3 : Imaginary
Concept:
If a, b, c are in AP, than \(b = \frac{{a + c}}{2}\)
If a, b, c are in GP than \(b = \sqrt {ac} \)
If a, b, c are in HP than \(b = \frac{{2ac}}{{a + c}}\)
Calculation:
assume a = 2, b = 8, n = 1
2, a1, 8 are in AP, than \({a_1} = \frac{{2 + 8}}{2} = 5\)
2, b1, 8 are in GP, than \({b_1} = \sqrt {2 \times 8} = 4\)
2, c1, 8 are in HP, than \({c_1} = \frac{{2 \times 2 \times 8}}{{2 + 8}} = \frac{{16}}{5}\)
Here, we have to find the nature of roots of the equation anx2 - bnx + cn = 0
By substituting n = 1 in the equation anx2 - bnx + cn = 0 we get
⇒ a1x2 - b1x + c1 = 0
We know that, a1 = 5, b1 = 4 and c1 = 16/5
So, the discrimination \(D = {b_1}^2 - 4{a_1}{c_1}\) for the equation a1x2 - b1x + c1 = 0 is \(D = {4^2} - 4 \times 5 \times \frac{{16}}{5} = - 48\)
∵ D < 0 so the roots are imaginary
Hence, option C is the correct answer.