A leprechaun places a magic penny under a girl\'s pillow. The next night there a

Question

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 $10billion? (Proceed by trial and error.)

Explanation / Answer

we need to know after howmany days the number of pennies reach the total value of Dollar 10 Billion.

That is the total number of pennies should be 1000 Billion = 1000000000000

After one night total number of pennies = 2

after two nights total number of pennies = 4

It will follow like this.

2,4,8,16,32,64,128,256,............

It can be observed that the nth term of the above sequence is 2n. n is natural number , and nth term gives the number of pennies after n nights.

after 30 nights, there will be 230 =   1,073,741,824 pennies under pillow   but its less than 1000 Billion

After 40 nights, there will be 240 = 1,099,511,627,776    But it goes higher than 1000 Billion

After 39 nights there will be 239 = 549,755,813,888 which is less than 1000 Billion

So after 40 nights the number of coins turns more than 1000 Billion in number for the first time.

Thus There will be 40 days elapsed before the girl has a total value of more than 10 Billion Dollars.

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