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Step-by-Step Solution
Step 1: Recognize the complex number α
We are given
$ \alpha = \frac{-1 + i\sqrt{3}}{2} $.
Notice that this is one of the cube roots of unity, often denoted by $ \omega $, where $ \omega^3 = 1 $ and $ 1 + \omega + \omega^2 = 0 $.
Step 2: Express the sum for a
We define:
$ a = \bigl(1 + \alpha\bigr) \sum_{k=0}^{100} \alpha^{2k}. $
Substitute $ \alpha = \omega $ to get:
$ a = \bigl(1 + \omega\bigr) \sum_{k=0}^{100} \omega^{2k}. $
Step 3: Simplify the series for a
The sum of a geometric series $ \sum_{k=0}^{n} r^k $ is:
$ \frac{1 - r^{n+1}}{1 - r},\;\text{provided }r \neq 1. $
Here, $ r = \omega^2 $, so
$ \sum_{k=0}^{100} \omega^{2k} = \frac{1 - \bigl(\omega^2\bigr)^{101}}{1 - \omega^2}. $
Hence:
$ a = \bigl(1 + \omega\bigr)\,\frac{1 - \omega^{202}}{1 - \omega^2}. $
Step 4: Further simplification using properties of ω
Since $ \omega^3 = 1, $ we can write:
$ \omega^{202} = \omega^{3 \cdot 67 + 1} = \bigl(\omega^3\bigr)^{67} \cdot \omega^1 = 1^{67} \cdot \omega = \omega. $
So,
$ 1 - \omega^{202} = 1 - \omega. $
Also note that:
$ 1 - \omega^2 = (1 - \omega)(1 + \omega). $
Putting these together:
$ a = \frac{(1 + \omega)\bigl(1 - \omega\bigr)}{(1 - \omega)(1 + \omega)} = 1. $
Step 5: Express the sum for b
We define:
$ b = \sum_{k=0}^{100} \alpha^{3k}. $
Again, substitute $ \alpha = \omega $:
$ b = \sum_{k=0}^{100} \omega^{3k}. $
Step 6: Simplify the series for b
Since $ \omega^3 = 1, $ then $ \omega^{3k} = 1^k = 1 $ for all integers k. Thus,
$ b = \sum_{k=0}^{100} 1 = 101. $
Step 7: Identify the quadratic equation
We have found that the two values (roots) are
$ a = 1 $ and $ b = 101. $
A quadratic equation with roots $ \alpha_1 $ and $ \alpha_2 $ is generally:
$ x^2 - (\alpha_1 + \alpha_2)x + \alpha_1 \alpha_2 = 0. $
Here, $ \alpha_1 = 1 $ and $ \alpha_2 = 101 $, so the sum of roots is $ 1 + 101 = 102 $, and the product of roots is $ 1 \times 101 = 101. $
Therefore, the required quadratic equation is:
$ x^2 - 102x + 101 = 0. $
Answer
The correct quadratic equation satisfying the conditions is
$ x^2 - 102x + 101 = 0. $
