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Question #40711

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha$ and $\beta$ be the roots of the equation $z^2 - \sqrt{2}z + 1 = 0$ . Let $z_0 = \frac{\sqrt{3} + i}{2}$ . A sequence is defined as $T_n = (z_0 \alpha)^n + (\overline{z_0} \beta)^n$ for $n \in \mathbb{N}$ . Find the value of $\sum_{n=1}^{95} (T_n^2 + T_n)$ .

Options

A.
186
B.
188
C.
190
D.
192

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Quadratic Roots in Sequences|Euler's Form and Rotations|Conjugate Root Theorem

Why This Question Is Tricky

Common Mistakes

This question tests the application of Euler's form and the Conjugate Root Theorem within sequence summations.

The primary trap lies in the ambiguity of assigning the roots $\alpha$ and $\beta$ .

Students often waste time trying to evaluate both cases separately or attempt a binomial expansion of the terms.

The conceptual leap is recognizing that multiplying by $z_0$ and $\overline{z_0}$ preserves the conjugate nature of the roots, ensuring $T_n$ is always a real, oscillatory sequence ( $2\cos(n\theta)$ ).

Furthermore, evaluating the sum by extending it to a full period (96 terms) and subtracting the 96th term is a classic JEE Advanced time-saving heuristic.

Failing to use Euler's form leads to an intractable algebraic mess.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Quadratic Roots in Sequences
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free