IPMAT Indore 2023 · Quantitative Ability · Question 29

Step 1 — track the directions. Starting east and turning right each time cycles east → south → west → north, repeating with period 4. The distances halve each leg: 32,16,8,4,2,1,\ldots

Step 2 — sum the east–west (x) displacements. East positive, west negative:

x=32-8+2-\frac{1}{2}+\cdots

a GP with first term 32 and ratio -1/4:

x=\frac{32}{1-\left( -1/4 \right)}=\frac{32}{5/4}=\frac{128}{5}

Step 3 — sum the north–south (y) displacements. South negative, north positive:

y=-16+4-1+\cdots =\frac{-16}{1+1/4}=-\frac{64}{5}

Step 4 — take the distance from the origin.

d=\sqrt{{x}^{2}+{y}^{2}}=\sqrt{{\left( \frac{128}{5}\right)}^{2}+{\left( \frac{64}{5}\right)}^{2}}=\frac{64}{5}\sqrt{{2}^{2}+{1}^{2}}=\frac{64\sqrt{5}}{5}=\frac{64}{\sqrt{5}}

Answer: Option 4 — \frac{64}{\sqrt{5}} meters

Shortcut. Group by axis, and use the period-4 structure. Legs 1 and 3 are opposite, as are 2 and 4 — so each axis gets its own GP with ratio -1/4 (two halvings plus a sign flip).

x=\frac{32}{1+1/4}=\frac{128}{5}, y=\frac{-16}{1+1/4}=-\frac{64}{5}

Note x=2\left| y \right| always, because the first leg is twice the second and everything after scales identically. So

d=\left| y \right|\sqrt{5}=\frac{64}{5}\sqrt{5}=\frac{64}{\sqrt{5}}

The general spiral rule: with halving steps and quarter turns, each axis collapses to \frac{\text{first leg on that axis}}{1+{r}^{2}} with r=1/2. That converts a four-series problem into two divisions.