Before You Begin

Question #40670

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{k=1}^n \frac{k^2+k-1}{(k^2+1)(k^2+2k+2)}$ . Find the value of $\sum_{n=1}^{50} \left[ 10 S_n \right]$ where $[x]$ denotes the greatest integer less than or equal to $x$ .

Options

A.
188
B.
200
C.
186
D.
192

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

The Telescopic Series Trap|Partial Fractions Decomposition|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

The problem tests the student's ability to identify a hidden telescopic series via partial fraction decomposition and subsequently evaluate its behavior inside a Greatest Integer Function (GIF).

The core trap lies in the asymptotic behavior of the telescopic sum.

Students often calculate the limit as $n \to \infty$ (which is $1/2$ ), multiply by 10 to get 5, and incorrectly assume the GIF evaluates to 4 or 5 uniformly for all terms.

By forcing a manual check of the boundary conditions where the fractional part crosses integer thresholds (specifically at $n=1, 2, 4,$ and $9$ ), the question filters out those who rely on blind limit approximations instead of rigorous discrete inequality analysis.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	The Telescopic Series Trap
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free