My new wrist-watch makes a beep sound, in every hour, as many as the time it…

Step-by-Step Solution

To find the total number of beeps in a day (24 hours), we need to calculate how many beeps occur in a 12-hour cycle and then multiply that by 2.

1. Understand the pattern:

   • The watch beeps according to the time it shows (e.g., at 1 o'clock, 1 beep; at 2 o'clock, 2 beeps; and so on).

   • In a 12-hour clock, the hours range from 1 to 12.

2. Calculate beeps for one 12-hour cycle:

   • Number of beeps = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12

   • This is the sum of an arithmetic series: n(n+1) / 2

   • Sum = 12 * (12 + 1) / 2 = 12 * 13 / 2 = 6 * 13 = 78 beeps.

3. Calculate beeps for a full day (24 hours):

   • A day has two 12-hour cycles.

   • Total beeps = 78 * 2 = 156 beeps.