I have math packet and have been stuck on this math packet all summer. I just want the formula to solve it.?

Question

"Snap your fingers. Wait one second and snap them again. Then wait 2 seconds and snap them again. Following this pattern, the next snap will come 4 seconds after that, than 8, 16 and so on. Each time, the number of seconds between snaps is doubled. If you follow this pattern carefully, how many times times will you snap your fingers in one year? Explain your reasoning."

Answer 1

Firstly, the number of seconds in one year is 365.25 days/year * 24 hours/day * 60 minutes/hour * 60 seconds/minute = 31,557,600 seconds.

The first finger snap sets time t = 0 seconds. The second snap has the interval 1 second (which is 2^0), and total time is 0 + 1 = 1 second. The third snap has the interval 2 seconds (which is 2^1), and the total time is 1 + 2 = 3 seconds. The fourth snap has the interval 4 seconds (which is 2^2), and the total time is 3 + 4 = 7 seconds. We can see the pattern is:

Interval time = 2^(# snaps - 2),

Total time = 1 + ∑ 2^(# snaps - 2) seconds.

Therefore, at the 26th snap, the time interval is 2^24 = 16,777,216 seconds, and the total elapsed time is 33,554,431 seconds (or 1.06 years). Strictly speaking then, the fingers were snapped 25 times in less than one year, with the last snap being about 6.5 months into the year. These power and sum calculations are easy using a spreadsheet like MS Excel.