Consider the quadratic equation \(\left({n}^{2}-2n+2\right){x}^{2}-3x+{\left({n}^{2}-2n+2\right)}^{2}=0,n\in R\). Let \(…

We have \(\left(n^2-2 n+2\right) x^2-3 x+\left(n^2-2 n+2\right)^2=0\)

Since \(n^2-2 n+2=(n-1)^2+1\)

Sum of roots \(=\frac{3}{(n-1)^2+1}\)

Since the denominator is minimum when \((n-1)^2=0\)

Therefore \(\beta=3\)

Product of roots \(=(n-1)^2+1\)

So the minimum value of the product is clearly

\(\alpha=1\)

Now, first term \(a=\alpha=1\)

Common ratio \(r=\frac{\alpha}{\beta}=\frac{1}{3}\)

Now, \(S_6=1 \cdot \frac{1-\left(\frac{1}{3}\right)^6}{1-\frac{1}{3}}\)

\(S_6=\frac{1092}{729}=\frac{364}{243}\)