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Question #41855

Difficulty: 8/10 | 🟢 Free

The Question

Evaluate the sum $\sum_{k=1}^{\infty} \frac{|k^2 - 11k + 28|}{2^{k+1}}$

Options

A.
$\frac{195}{32}$
B.
$6$
C.
$\frac{189}{32}$
D.
$\frac{99}{16}$

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

Algebra of Functions|Sigma Notation Properties|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This problem tests the student's ability to handle the summation of a sequence involving a modulus where the underlying sequence changes sign.

The core trap lies in blindly applying the method of differences (telescoping sum) or Arithmetico-Geometric Progression (AGP) formulas without splitting the summation domain at the roots of the quadratic numerator ( $k=4$ and $k=7$ ).

If the modulus is ignored, the sum incorrectly telescopes to simply $V_1 = 6$ .

By forcing the student to identify the sign change, the problem evaluates their rigor in algebraic reduction and domain constraints.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Relations and Functions - Class 11
Concept	Algebra of Functions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free