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Question #41420

Difficulty: 8/10 | 🟢 Free

The Question

Let $A_r = \sum_{k=0}^r \binom{n}{k} \binom{n-k}{r-k}$ for $0 \le r \le n$ . For an integer $n \ge 2$ , define the function $f_n(x)$ for $x > -\frac{1}{2}$ by $f_n(x) = \int_0^x \frac{\sum_{r=2}^n A_r t^r}{t^2(2t+1)} dt$ Let $g_n(x) = f_n(x) - 2(n-1)\ln(2x+1)$ . Let $S$ be the set of all integers $n \ge 2$ such that $g_n(x)$ is a constant function on $\left(-\frac{1}{2}, \infty\right)$ . Find the value of $\sum_{n \in S} n + g_4(1)$ .

Options

A.
14
B.
18
C.
20
D.
24

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

Partial Fractions Decomposition|Properties of Binomial Coefficients|Coefficient of x^r in Expansion|Constant Function Logic

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to extract a polynomial from a complex binomial sum and perform advanced partial fraction decomposition.

The primary trap is the 'Constant Function Logic' distractor.

Students often assume that subtracting the logarithmic term $2(n-1)\ln(2x+1)$ from the integral leaves a constant zero function, mistakenly thinking the numerator fully cancels out.

However, they miss the 'binomial residue'—the remaining polynomial $\frac{(1+2x)^{n-1} - 1 - 2(n-1)x}{x^2}$ which only vanishes for $n=2$ .

Recognizing this residue is crucial to correctly identifying the set $S$ and evaluating $g_4(1)$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Integrals - Class 12
Concept	Partial Fractions Decomposition
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free