A leprechaun places a magic penny under a girl's pillow. The next night there are 2 magic pennies under her pillow. The following morning she finds four pennies. Apparently, while she sleeps each penny turns into two magic pennies. The total number of pennies seen under the pillow each day is the grand total; that is, the pennies from each of the previous days are not being stored away until more pennies magically appear. How many days would elapse before she has a total of more than $5 billion ? (Proceed by trial and error.)

nothing days

(Type a whole number.)

34

Step-by-step explanation:

Since the pennies need to be kept under the pillow in order to be doubled, we're really looking for the day the girl will wake up with 5 billion pennies under her pillow.

The general formula of that geometric sequence will be: t = 1 + 2^ (n-1)

We need to find the value of n when t will be over 5,000,000,000.

So, the equation becomes:

5,000,000,001 = 1 + 2^ (n-1)

2^ (n-1) = 5,000,000,000

log (2^ (n-1)) = log (5,000,000,000)

(n-1) * log (2) = log (5,000,000,000)

n-1 = log (5,000,000,000) / log (2)

n-1 = 22.25

n = 33.25

Since it has to be complete days, and we have to reach over 5 billion, then we round it up to 34.