A clock goes slow from midnight by 5 minutes at the end of the first hour, by 10 minutes at the end of the second hour, by 15 minutes at the end of the third hour and so on. What will be the time by this clock after 6 hours?

### Concept & Arithmetic Progression The clock is losing time in a linear pattern that corresponds directly to the number of elapsed hours. At the end of hour $n$, the clock is slow by $5 \times n$ minutes. ### Step-by-Step Solution 1. **Identify the pattern of delay:** - End of 1st hour: 5 minutes slow - End of 2nd hour: 10 minutes slow - End of 3rd hour: 15 minutes slow The delay at the end of the $n$-th hour is $5n$ minutes. 2. **Calculate the delay after 6 hours:** Let $n = 6$. Total delay = $5 \times 6 = 30 \text{ minutes}$. 3. **Determine the displayed time:** The actual elapsed time from midnight is 6 hours, meaning the true time is 6:00 a.m. Since the clock is 30 minutes slow, we subtract this delay from the true time. Displayed time = 6:00 a.m. - 30 minutes = 5:30 a.m. ### Exam Strategy & Shortcut Recognize immediately that the delay is simply $5 \times \text{hours}$. For 6 hours, it is 30 minutes slow. 6:00 a.m. actual time minus 30 minutes gives 5:30 a.m. ### Common Pitfall Confusing "goes slow" with "goes fast" and adding the time (resulting in 6:30 a.m.) instead of subtracting it. ### Final Answer Therefore, the correct answer is **5.30 a.m.**