Final answer:
By using the formula for the sum of an arithmetic series, the total number of buzzes a clock makes in a 24-hour period is calculated as 156. This is obtained by summing the buzzes for each of the 12 hours in a half-day and then doubling it for the full day. Therefore, the correct answer would be (b) 156.
Step-by-step explanation:
The question you've asked is related to a pattern of buzzes made by a clock over a 24-hour period. To find the total number of buzzes in a day, we should sum the number of buzzes for each hour. The clock buzzes once at 1 O'clock, twice at 2 O'clock, and so on, up to 12 times at 12 O'clock. This pattern then repeats for the next 12 hours.
Let's split the day into two halves of 12 hours each:
- For the first 12 hours (1 AM to 12 PM), the total buzzes are 1+2+3+...+12.
- The second 12 hours (1 PM to 12 AM) will have the same number of buzzes as the first 12 hours.
To calculate the sum for 12 hours, we use the formula for the sum of an arithmetic series:
S = n/2 * (a1 + an)
Where:
- n is the number of terms
- a1 is the first term
- an is the last term
So, for the first 12 hours:
S = 12/2 * (1 + 12) = 6 * 13 = 78
Since the pattern repeats, we multiply this sum by 2 to cover the full 24 hours.
Total buzzes = 78 * 2 = 156
Therefore, the correct answer would be (b) 156.