Sum of GP Terms from Quadratic Equation Parameters

To find the sum of the first six terms of the given Geometric Progression (G.P.), we first need to determine the values of $\alpha$ and $\beta$.

Step 1: Simplify the quadratic equation The given quadratic equation is: $(n^2 - 2n + 2)x^2 - 3x + (n^2 - 2n + 2)^2 = 0, \quad n \in \mathbf{R}$

Let us complete the square for $n^2 - 2n + 2$: $n^2 - 2n + 2 = (n - 1)^2 + 1$ Since $(n - 1)^2 \ge 0$ for all real $n$, we have: $n^2 - 2n + 2 \ge 1$

Let $k = n^2 - 2n + 2$, where $k \ge 1$. The quadratic equation can now be written as: $k x^2 - 3x + k^2 = 0$

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Step 2: Find the product and sum of the roots For the quadratic equation $k x^2 - 3x + k^2 = 0$:

• The product of the roots is: $P(k) = \frac{k^2}{k} = k$
• The sum of the roots is: $S(k) = \frac{3}{k}$

To ensure real roots exist, the discriminant $D$ must be non-negative: $D = (-3)^2 - 4(k)(k^2) = 9 - 4k^3 \ge 0 \implies k \le \left(\frac{9}{4}\right)^{1/3} \approx 1.31$ Since $k \ge 1$, the valid range for $k$ is $1 \le k \le \left(\frac{9}{4}\right)^{1/3}$.

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Step 3: Determine $\alpha$ and $\beta$

• $\alpha$ is the minimum value of the product of roots, $P(k) = k$. Since $k \ge 1$, the minimum value occurs at $k = 1$: $\alpha = 1$

• $\beta$ is the maximum value of the sum of roots, $S(k) = \frac{3}{k}$. Since $S(k)$ is a decreasing function for $k > 0$, its maximum value also occurs at the smallest allowed value of $k$, which is $k = 1$: $\beta = \frac{3}{1} = 3$

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Step 4: Calculate the sum of the first six terms of the G.P. For the given G.P.:

• First term $a = \alpha = 1$
• Common ratio $r = \frac{\alpha}{\beta} = \frac{1}{3}$

The formula for the sum of the first $n$ terms of a G.P. is: $S_n = \frac{a(1 - r^n)}{1 - r}$

For $n = 6$: $S_6 = \frac{1 \cdot \left(1 - \left(\frac{1}{3}\right)^6\right)}{1 - \frac{1}{3}}$

$S_6 = \frac{1 - \frac{1}{729}}{\frac{2}{3}} = \frac{\frac{728}{729}}{\frac{2}{3}} = \frac{728}{729} \times \frac{3}{2} = \frac{364}{243}$

Thus, the sum of the first six terms of the G.P. is $\frac{364}{243}$.

Correct Option: C