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GATE 2025 XH2 (English) – Question 4

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

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

Note: The figures shown are representative.

Iteration 0 is a straight line of length 1. Iteration 1 replaces it by a curve made of 5 segments, each of length 1/3. Iteration 2 replaces every segment of iteration 1 in the same way, giving 25 segments, each of length 1/9.
  1. $\left(\frac{5}{3}\right)^{\frac{n}{2}}$
  2. $\left(\frac{5}{3}\right)^{n}$
  3. $\left(\frac{5}{3}\right)^{2n}$
  4. $\left(\frac{5}{3}\right)^{n(2n-1)}$

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Correct answer: (B) $\left(\frac{5}{3}\right)^{n}$

Explanation

Each step replaces every segment by 5 segments that are one third as long. After iteration $n$ there are $5^n$ segments, each of length $\left(\frac{1}{3}\right)^n$, so the total length is $\left(\frac{5}{3}\right)^n$. This agrees with iteration 1 ($5 \times \frac{1}{3} = \frac{5}{3}$) and iteration 2 ($25 \times \frac{1}{9} = \frac{25}{9}$).