Quadratics and Progressions

Algebra Level 5

For constant \(a\), let the roots (of \(x\)) of the equation \(x^2+ 2(a-3) x+9= 0 \) lie between \(-6\) and \(1\).

Given that

\(2, h_1, h_2, h_3,\ldots,h_{20} , \lfloor a\rfloor \) follows a harmonic progression.
\(2, a_1, a_2, a_3,\ldots,a_{20} , \lfloor a\rfloor \) follows an aritmetic progression.

Find the value of \(a_3 \times h_{18} \).

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