Question

The following figures show three curves generated using an iterative algorithm. The total length of the curve generated after 'Iteration n' is: 

 

• A. \( \left( \frac{5}{3} \right)^{\frac{n}{2}} \)
• B. \( \left( \frac{5}{3} \right)^n \)
• C. \( 2n \)
• D. \( \left( \frac{5}{3} \right)^n(2n - 1) \)

Hint:

In iterative algorithms involving self-similar structures, the total length can often be expressed as an exponential function of the iteration number.

Answer 1

The Correct Option is B

Step 1: Analyzing the iterative process.
In the first iteration (Iteration 0), the length of the curve is 1. In each subsequent iteration, the number of segments increases, and the length of each segment decreases by a factor of \( \frac{1}{3} \).

Step 2: Finding the total length after each iteration.
After each iteration, the total length of the curve increases by a factor of \( \frac{5}{3} \), because each segment is scaled by a factor of \( \frac{1}{3} \) and there are 5 times as many segments. Thus, the total length after 'Iteration n' is:
\[ \text{Total length} = \left( \frac{5}{3} \right)^n. \]