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Question #40704

Difficulty: 8/10 | 🟢 Free

The Question

Let $S_n = \sum_{r=1}^n r^3$ . Consider the function $f(n) = \frac{S_n}{n^2} + \frac{32}{n+1}$ defined for all $n \in \mathbb{N}$ . Find the minimum value of $f(n)$ subject to the condition $|n^2 - 3n - 2| \ge 26$ .

Options

A.
12
B.
16
C.
20
D.
24

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Sum of Powers of Natural Numbers|Arithmetic Mean Inequality|Modulus Inequality Logic

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of optimization via the AM-GM inequality with discrete domain restrictions.

The primary trap lies in the unconstrained optimization: applying AM-GM directly yields a minimum value of 12 at n=3.

Students who blindly apply the inequality without verifying the secondary modulus constraint will select the trap option (A).

The modulus inequality subtly restricts the domain to n ≥ 7, invalidating the AM-GM equality condition.

This forces a piecewise analysis where the student must recognize that the function is strictly increasing for n ≥ 3, making the boundary value n=7 the true constrained minimum.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Sequences and Series - Class 11
Concept	Sum of Powers of Natural Numbers
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free