GATE 2025 EC GA Q.4: solution and answer

Official answer: BSource: Official answer key, GATE 2025 EC (IIT Roorkee)

1. Read the rule from Iteration 1. The straight segment is cut into thirds, and the middle third is replaced by three sides of a square (up, across, down), each side $\frac{1}{3}$ long. So one segment becomes 5 segments, each $\frac{1}{3}$ as long.

2. Iteration 0: 1 segment of length 1, so $L_0=1$.

3. Iteration 1: $5$ segments of length $\frac{1}{3}$, so $L_1=5\times \frac{1}{3}=\frac{5}{3}$.

4. Iteration 2: each of those 5 segments is replaced by the same motif at one-third scale. That gives $5^2=25$ segments of length $\frac{1}{9}$, which you can count in the figure: 5 on each baseline piece, 5 on each vertical side and 5 on the top. So $L_2=25\times \frac{1}{9}=\frac{25}{9}={\left(\frac{5}{3}\right)}^2$. The direction each small bump points in does not change the length. This is why the paper says the figures are representative.

5. General term: each iteration multiplies the number of segments by 5 and divides the length of each segment by 3. So the total length is multiplied by $\frac{5}{3}$ every time: $$L_n=5^n\times {\left(\frac{1}{3}\right)}^n={\left(\frac{5}{3}\right)}^n$$

6. Check the options against $L_0=1$, $L_1=\frac{5}{3}$ and $L_2=\frac{25}{9}$. Option (B) gives $1$, $\frac{5}{3}$ and $\frac{25}{9}$, so all three match. Option (A) gives $\sqrt{5/3}$ at $n=1$ and option (C) gives ${\left(\frac{5}{3}\right)}^2$ at $n=1$, so both fail. Option (D) gives ${\left(\frac{5}{3}\right)}^1=\frac{5}{3}$ at $n=1$, which matches, but at $n=2$ it gives ${\left(\frac{5}{3}\right)}^{2\times 3}={\left(\frac{5}{3}\right)}^6$, which does not. Only (B) fits every iteration shown.

7. Answer: (B) ${\left(\frac{5}{3}\right)}^n$