A bus starts running with an initial speed of 21 km/h, and its speed increases every hour by 3 km/h. How many hours will it take to cover a distance of 252 km?

Introduction / Context:
This is a time and distance problem involving a bus whose speed does not remain constant. Instead, the bus starts at a given initial speed and increases its speed by a fixed amount every hour. We must calculate how long it takes to cover a known distance. This is essentially a question about the sum of distances covered in consecutive hours with speeds forming an arithmetic progression.

Given Data / Assumptions:
- Initial speed of the bus = 21 km/h in the first hour.
- Every hour the speed increases by 3 km/h.
- Total distance to be covered = 252 km.
- The change in speed is stepwise at the end of each hour, and within each hour the speed is constant.

Concept / Approach:
The speed in each hour forms an arithmetic sequence: 21 km/h in the first hour, 24 km/h in the second hour, 27 km/h in the third hour, and so on, with common difference 3 km/h. The distance covered in each hour is simply the speed for that hour, because time per hour is 1 hour. Thus, the total distance covered in n hours is the sum of the first n terms of that arithmetic sequence. We use the formula for the sum of an arithmetic series and then equate it to 252 km to solve for n.

Step-by-Step Solution:
Step 1: Write the sequence of speeds by hour:Hour 1: 21 km/h, Hour 2: 24 km/h, Hour 3: 27 km/h, etc.Step 2: Let n be the number of hours required to cover 252 km.Speed in the k-th hour = 21 + 3(k - 1).Step 3: Distance covered in each hour = speed (since time is 1 hour per term).Total distance in n hours = sum of first n terms of the arithmetic sequence:S_n = n/2 * [2 * 21 + (n - 1) * 3].Step 4: Set S_n = 252 and solve for n:n/2 * [42 + 3(n - 1)] = 252.Simplify: 42 + 3(n - 1) = 42 + 3n - 3 = 39 + 3n.So S_n = n/2 * (39 + 3n) = (3n(13 + n)) / 2 = 252.Step 5: Multiply both sides by 2/3:n(13 + n) = 168.This gives n^2 + 13n - 168 = 0.Step 6: Factor or use quadratic formula; the equation factors as (n - 8)(n + 21) = 0, so n = 8 (positive root).

Verification / Alternative check:
Compute the distance covered in 8 hours explicitly: speeds are 21, 24, 27, 30, 33, 36, 39, 42 km/h. Their sum is (21 + 42) * 8 / 2 = 63 * 4 = 252 km. This confirms that the bus covers exactly 252 km in 8 hours. Therefore, 8 is the required number of hours.

Why Other Options Are Wrong:
- 3, 5, and 10 hours, when plugged into the arithmetic series sum formula, yield total distances different from 252 km. For example, 5 hours gives a much smaller total distance, while 10 hours gives a larger distance than 252 km.

Common Pitfalls:
Some students treat the speed as increasing continuously instead of in hourly steps, or they mistakenly assume the speed remains fixed at the final value. Others forget to use the arithmetic progression formula and instead try to sum incorrectly. Writing out the pattern of speeds clearly and using the series sum formula helps avoid confusion and ensures accuracy.

Final Answer:
The bus will take 8 hours to cover 252 km.