Before You Begin

Question #40974

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x)$ be the sum of the first 20 terms of the series $1 + 4(x-2) + 7(x-2)^2 + 10(x-2)^3 + \dots$ If $R$ is the range of $f(x)$ for $x \in [2, 3]$ , find the coefficient of range of $R$ .

Options

A.
$\frac{589}{591}$
B.
1
C.
$\frac{1179}{1181}$
D.
$\frac{58}{61}$

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

Range and Coefficient of Range|Arithmetic-Geometric Progression (AGP)|The Zero-Factor Cancellation Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Zero-Factor Cancellation Trap' within the context of an Arithmetic-Geometric Progression (AGP).

Students typically use the standard AGP sum formula, which involves dividing by $(1-r)^2$ .

When evaluating the maximum of the range at $x=3$ (where $r=1$ ), the formula yields a $0/0$ indeterminate form.

Students who blindly trust the formula often assume the function has a vertical asymptote (leading to an incorrect coefficient of range of 1) or make errors applying L'Hopital's rule (often forgetting the denominator's second derivative, leading to option C).

The mathematically sound approach is to recognize that at $r=1$ , the sequence collapses into a simple Arithmetic Progression, bypassing the singularity entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Statistics - Class 11
Concept	Range and Coefficient of Range
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free