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Question #40690

Difficulty: 8/10 | 🟢 Free

The Question

Let the sequence of points $P_n(x_n, y_n)$ be defined by $x_n = \left( \sum_{k=1}^n \frac{1}{k(k+1)} \right)^2 \quad \text{and} \quad y_n = 2 \sum_{k=1}^n \frac{1}{k(k+1)}$ for $n \in \mathbb{N}$ . Let $L_n$ be the straight line passing through $P_n$ and $P_{n+1}$ . Let $S$ be the set of all $n \in \mathbb{N}$ such that the origin $(0,0)$ and the point $A(15, 17)$ lie on the same side of the line $L_n$ . Find the sum of all elements in $S$ .

Options

A.
78
B.
91
C.
105
D.
120

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Method of Differences|Parametric form pitfalls|Point-Line Relative Position

Why This Question Is Tricky

Common Mistakes

This question tests the synergy between the Method of Differences and parametric coordinate geometry.

The primary trap is algebraic brute-force: students who fail to recognize that $P_n$ lies on the parabola $y^2 = 4x$ with parameter $t_n$ will attempt to find the line equation using messy rational coordinates, leading to severe time loss and calculation errors.

Furthermore, the relative position constraint requires evaluating the sign of the line's expression at the origin and $A(15,17)$ .

A common pitfall is incorrectly assuming the line's constant term is positive without checking $F_n(0,0)$ , or mishandling the strict inequality $F_n(15,17) > 0$ , which tests precise quadratic root estimation to identify the exact integer boundary.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Method of Differences
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free