Question
If $$a,{a_1},{a_2},{a_3},.....,{a_{2n - 1}},b$$ are in A.P., $$a,{b_1},{b_2},{b_3},.....,{b_{2n - 1}},b$$ are in G.P. and $$a,{c_1},{c_2},{c_3},.....,{c_{2n - 1}},b$$ are in H.P., 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 :
imaginary
Solution :
Clearly $${a_n},{b_n},{c_n}$$ are the middle terms of the given A.P., G.P., H.P.
respectively. So, $${a_n}$$ is the AM of $$a,b;{b_n}$$ is the GM of $$a, b$$ and $${c_n}$$ is the HM of $$a, b.$$ Also $${a_n},{b_n},{c_n}$$ are positive because $$a, b$$ are positive.
∴ $${a_n},{b_n},{c_n}$$ are in G.P.; so discriminant $$ = b_n^2 - 4{a_n}{c_n} = - 3{a_n}{c_n} < 0.$$