Limits, Continuity & Differentiability
Limits of Sequences
Grade 12
Question:
<p>A circular disk of unit radius is filled with a number of smaller circular disks arranged in the form of hexagon. Let \(A_n\) denotes a stack of disks arranged in the shape of a hexagon having \(n\) disks on a side. If \(A\) be the area of large disk, \(S_n\) be the number of disks in \(A_n\) configuration and \(r_n\) be the radius of each disk in \(A_n\) configuration, then find \(\lim_{n \to \infty} \frac{S_n}{n^2}\)</p>
<p>(a) 3</p>
<p>(b) 4</p>
<p>(c) 1</p>
<p>(d) \(\frac{11}{3}\)</p>
Step-by-Step Solution
Key Concept: Recognize the hexagonal packing pattern and that the number of disks grows as $O(n^2)$, making the limit equal to the leading coefficient.
<p>For a hexagonal arrangement of disks with $n$ disks on a side, the total number of disks follows a pattern related to hexagonal numbers. The number of disks in configuration an is $S_n = 3n^2 - 3n + 1$. Therefore, $$\lim_{n \to \infty} \frac{S_n}{n^2} = \lim_{n \to \infty} \frac{3n^2 - 3n + 1}{n^2} = 3$$</p>
Correct Answer: a