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 ​$
1 trillion

Answer 1

Answer:

31

Step-by-step explanation:

On the first day there was 1 penny under the pillow.  

If each penny become 2 times and the total number of pennies become (x+2x)=3x {Where x is the number of pennies in the previous day}

Therefore, the total number of pennies in the 2nd day =3×1 = 3

Similarly, the total number of pennies in the 3rd day =3×3 = 9 and the total number of pennies in the 4th day = 3×9 =27 and so on.

Therefore, the numbers of pennies are increasing in the series 1,3,3²,3³, .... and so on.

Now, $1 trillion =100 trillion pennies =$10^{2}*10^{12}$ pennies =$10^{14}$ pennies.

Let us assume that, after n days the number of pennies will be greater than $10^{14}$ pennies.

Assume, $3^{n-1}=10^{14}$ {Taking log on both sides we get the following}

⇒(n-1)㏒3=14 ㏒10=14

⇒n-1 =$\frac{14}{log3}$= 29.34

⇒ n = 30.34.

Therefore. n has to be an integer greater than 30.34 i.e. 31.

So, on the 31st day, the number of pennies will be more than $1 trillion.

(Answer)

Answer 2

Answer:

41st day she has a total more than  ​$ 1 trillion

40 days will elapse

Step-by-step explanation:

First day Pennies = 1

Second day total Pennies = 2

Third Day total pennies = 4

and son

on nth day Total pennies = 2ⁿ⁻¹

2ⁿ⁻¹  >  10¹²

=>( n - 1) log 2 > 12

=> n - 1 > 12/0.301

=> n - 1 > 39.87

=> n > 40.87

=> n = 41

on 41st day she has a total more than  ​$ 1 trillion

40 days will elapse