- Note that $\alpha +\beta = \dfrac{-c}{1} =-c$ and $\alpha\beta = \dfrac{-9}{1}=-9$.
$\begin{array}{cl}
& \alpha^2 +\beta^2 \\
= & \alpha^2 +2\alpha\beta +\beta^2 -2\alpha\beta \\
= & (\alpha+\beta)^2 -2\alpha\beta \\
= & (-c)^2 -2(-9) \\
= & c^2 +18
\end{array}$ - The common difference of the arithmetic
$\begin{array}{cl}
= & \alpha^2 +\beta^2 -c^2 \\
= & (c^2 +18) -c^2 \\
= & 18
\end{array}$Consider the $3$rd term, we have
$\begin{array}{rcl}
c^2 +18 +18 & = & 85 \\
c^2 & = & 49
\end{array}$Hence, we have
$\begin{array}{rcl}
S(n) & > & 2\times 10^6 \\
\dfrac{n}{2} [2(49) +(n-1)(18)] & > & 2 \times 10^6 \\
18n^2 +80n -4\times 10^6 & > & 0 \\
9n^2 +40n -2\times 10^6 & > & 0
\end{array}$Therefore, $n< \dfrac{-40-\sqrt{40^2-3(9)(-2\times 10^2)}}{2(9)}$ or $n> \dfrac{-40+\sqrt{40^2-3(9)(-2\times 10^2)}}{2(9)}$.
Therefore, $n< -473.631\ 980\ 8$ or $n> 469.187\ 536\ 4$.
Therefore, the least value of $n$ is $470$.
2022-I-17
Ans: (a) $c^2+18$ (b) $470$
