CAT 2022 Slot 1 · Quant · Question 12

Combine the two fractions inside the floor: \frac{1}{5}+\frac{n}{25}=\frac{5+n}{25}.

So each term is the largest integer not exceeding \frac{n+5}{25}.

That value is 0 while n+5<25, that is for n from 1 to 19 — nineteen terms contributing nothing.

It becomes 1 once n+5 reaches 25, that is from n=20 onwards, and stays 1 while n+5<50, that is up to n=44.

It becomes 2 from n=45 onwards.

So the running total stays at 0 through n=19, then climbs by 1 for each n from 20 up.

After N terms, with N in the range 20 to 44, the sum is N-19.

Setting N-19=25 gives N=44.

Checking the boundary: N=44 sits exactly at the top of that stretch, since the next term, at n=45, would add 2 rather than 1.

Shortcut. Most of the terms are zero, so only the point where the floor first turns 1 matters. That happens at n=20, and 25 ones take the index to 20+25-1=44.