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Question #40758

Difficulty: 8/10 | 🟢 Free

The Question

A sequence of points $P_1, P_2, P_3, \dots$ on the X-axis is constructed such that $P_1 \equiv (0,0)$ and $P_2 \equiv (10,0)$ . For every integer $n \ge 3$ , the point $P_n$ divides the directed line segment $P_{n-2}P_{n-1}$ externally in the ratio $1:3$ . If $x_n$ denotes the abscissa of $P_n$ , find $\lim_{n \to \infty} x_n$ .

Options

A.
4
B.
-5
C.
0
D.
\text{Does not exist}

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

Internal vs External Division|Insertion of Means|Infinite GP Convergence

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous application of limits to recursively defined coordinate sequences, specifically targeting the 'Infinite GP Convergence' trap.

Students frequently derive the common ratio of the coordinate differences and blindly apply the infinite geometric series sum formula $S = a/(1-r)$ to find the 'limit point' (yielding 4).

However, they fail to verify the domain constraint for convergence ( $|r| < 1$ ).

Because the external division ratio generates a common ratio of -3/2, the sequence of points diverges, rendering the limit non-existent.

This punishes rote memorization over fundamental convergence analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Internal vs External Division
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free