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Question #41136

Difficulty: 8/10 | 🟢 Free

The Question

Let $\alpha, \beta$ (where $\alpha < \beta$ ) be the roots of the equation $x^2 - 7x + 12 = 0$ . Let $f(x) = \alpha x - [\alpha x]$ and $g(x) = x + \frac{1}{\beta}$ , where $[t]$ denotes the greatest integer less than or equal to $t$ . Two sequences $\{x_n\}$ and $\{y_n\}$ are defined as follows: $x_1 = 0, \quad x_{n+1} = f(g(x_n)) \text{ for } n \ge 1$ $y_1 = 0, \quad y_{n+1} = g(f(y_n)) \text{ for } n \ge 1$ Find the value of $\sum_{n=1}^{120} (x_n + y_n)$ .

Options

A.
119
B.
178
C.
134
D.
120

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Composition Non-Commutativity|Quadratic Roots in Sequences|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the non-commutativity of function composition involving the greatest integer function (GIF) and its effect on recursive sequences.

The core trap lies in evaluating the nested GIF at integer boundaries.

Students often incorrectly assume that the fractional part of an integer is 1 instead of 0 (e.g., $\{3\} = 1$ ), or they assume $f(g(x))$ and $g(f(x))$ will converge to the same steady-state sequence.

By forcing the sequence through $x_3 = \{3\} = 0$ and $y_3 = 1 \implies y_4 = \{3\} + 1/4 = 1/4$ , the problem requires meticulous tracking of phase-shifted jump discontinuities.

Failing to reset the sequence at these integer boundaries leads to an incorrect periodic sum (yielding the trap answer 178), penalizing students who rely on smooth linear recurrence assumptions.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Composition Non-Commutativity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free