Before You Begin

Question #41125

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{N} \to \mathbb{Z}$ be defined by $f(n) = \left\lfloor \sum_{k=1}^n \log_3 \left( 1 + \frac{107}{k^2 + 108k} \right) \right\rfloor$ where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ . Let $S$ be the range of $f$ . Which of the following represents the cardinality of $S$ ?

Options

A.
4
B.
5
C.
6
D.
\text{Infinite}

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

Surjective Mapping (Onto)|The Telescopic Series Trap|Logarithmic Series

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to evaluate a telescoping logarithmic series and analyze its asymptotic boundary.

The primary trap is assuming the series diverges (since logarithmic sums often do, mimicking the harmonic series) leading to an 'Infinite' range.

A secondary trap is miscalculating the boundary limit $\log_3(108)$ or failing to check the initial value $f(1)=0$ , which leads to an off-by-one error (choosing 4 or 6).

The step-size analysis is crucial to guarantee surjectivity onto the finite integer set $\{0, 1, 2, 3, 4\}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 12
Concept	Surjective Mapping (Onto)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free