A clock sounds every half hour and every whole hour. Every half hour, it rings once. Every whole hour, it sounds a number of times based on the clock face as follows: the o’ clock that the clock shows is the number of times the clock sounds. For example, it sounds 8 times at 8 o’clock a.m., 2 times at 2 o’clock p.m., and 12 times at 12 o’clock p.m. How many times will it ring in a whole day?

times

The hour hand goes 2 times around the clock face in a whole day. Since a day has 24 half hours and 24 whole hours, the clock sounds 24 + (1 + 2 + 3 + … + 12) × 2 times.

To calculate the sum in parentheses, we use the formula for an arithmetic series. The terms of the sum consist of an arithmetic sequence with first term 1, last term 12, common difference 1, and number of terms 12. So the number of times is

24 + [ (1 + 12) × 12 ÷ 2] × 2 = 24 + 13 × 6 × 2 = 24 + 156 = 180.