# A leprechaun places a magic penny under a girl's pillow. The next night, there are two magic...

## Question:

A leprechaun places a magic penny under a girl's pillow. The next night, there are two 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 $2 trillion? (by trial and error method) ## Exponential Growth Exponential growth is the increase of a quantity at a certain rate or percentage over time. It is extensively used in modeling various systems in physics, biology, economics, and finance, such as the growth in the population of bacteria, nuclear chain reactions, and economic growth. ## Answer and Explanation: If the number of pennies doubles every night, then the increase would be exponential. Initially, one penny was found under her pillow. The number of pennies became 2 and 4, after 1 and 2 days, respectively. Suppose we let {eq}N {/eq} be the number of pennies after {eq}n {/eq} days, the equation would be given by $$N(n)=2^n\\[0.2cm]$$ Note that {eq}\$1=100\, \rm penny {/eq}. Hence, $2 trillion is equivalent to $$\2\times 10^{12}=2\times 10^{14}\,\rm pennies$$ Therefore, the number of days until she has a number of pennies that is greater than$2 trillion would be given by

\begin{align} 2\times 10^{14} & < 2^n\\[0.2cm] \ln \mid 2^n \mid &> \ln \mid 2\times 10^{14} \mid \\[0.2cm] n\ln \mid 2 \mid & > \ln \mid 2\times 10^{14} \mid \\[0.2cm] n &> \dfrac{\ln \mid 2\times 10^{14} \mid }{\mid 2 \mid}\\[0.2cm] n &> 47.5 \end{align}

Therefore, the girl would have a total number of pennies that is greater than \$2 trillion after 48 days.