Sequence of the form $pn^2+qn+r$

Question

A sequence 192, 360, 576 is formed by multiplying the corresponding two different arithmetic progression. What is the eighth term of the sequence?

Solution says that answer will be of the form $pn^2+qn+r$. Why is this, and how can I find it?

Hint: Try induction. – Ahaan S. Rungta Oct 31 '13 at 14:59 — Ahaan S. Rungta, Oct 31 '13 at 14:59

score 3 · Accepted Answer · answered Oct 31 '13 at 15:03

Let's let the first arithmetic progression be given by $x_n=a+nc$ and the second by $y_n=b+nd,$ so that your sequence is given by $$z_n=x_ny_n=bdn^2+(ad+bc)n+ab.$$ Put $r=ab,$ $q=ad+bc,$ and $p=bd$ for simplicity (since we aren't actually interested in the original sequences, anyway). In particular, we are given the following system: $$\begin{cases}192=r & \\360=p+q+r & \\576=4p+2q+r\end{cases}$$

Solving the above system for $p,q,$ and $r,$ what can we then do?

Sequence of the form $pn^2+qn+r$

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