Before You Begin

Question #41663

Difficulty: 8/10 | 🟢 Free

The Question

For each integer $n \ge 1$ , let $E_n$ be a plane whose intercepts $A_n, B_n, C_n$ on the $x, y, z$ axes respectively satisfy: $\frac{1}{A_n} = \sum_{r=0}^n \binom{n}{r}^2, \quad \frac{1}{B_n} = \sum_{r=0}^n r \binom{n}{r}^2, \quad \frac{1}{C_n} = \sum_{r=0}^n r^2 \binom{n}{r}^2$ Evaluate the sum: $\sum_{n=1}^{20} \left( 1 - \frac{A_n C_n}{B_n^2} \right)^{-1}$

Options

A.
420
B.
210
C.
840
D.
400

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

Variable plane intercepts|Vandermonde's Identity|Coefficient of x^r in Expansion|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to algebraically manipulate binomial sums using Vandermonde's Identity and its derivatives, framed within 3D geometry.

The primary trap lies in the non-linear dependency of the plane's intercepts.

Students often attempt to calculate the first few terms manually to guess a pattern, but calculation errors in $S_2$ (due to the $r^2$ term) are extremely common.

The rigorous path requires recognizing that $r$ and $r^2$ can be absorbed into the binomial coefficients via $r\binom{n}{r} = n\binom{n-1}{r-1}$ , transforming the complex sums into standard Vandermonde convolutions.

Failing to express all sums as scalar multiples of $\binom{2n}{n}$ leads to an algebraic dead end.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Variable plane intercepts
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free