Krista put 1 cent into her new bank on a Sunday morning. On Monday she put 2 cents into her bank. On Tuesday she put 4 cents into her bank, and she continued to double the amount of money she put into her bank each day for two weeks. On what day of the week did the total amount of money in her bank first exceed \($5\)?

tertre Jan 5, 2018

#1**+3 **

We are looking for this

.01 [ 1 - 2^N[ / [ 1 - 2] = 5

1 - 2^N = -500

-2^N = -501

2^N = 501 take the log of each side

log 2^N = log 501

N * log 2 = log 501

N = log501/log2

N = 8.9 ⇒ 9 days ⇒ [ Monday ]

CPhill Jan 5, 2018

#3**+2 **

.If the n days have passed since Sunday, then the total number of cents in her bank account is \(1+2+\cdots+2^n\). This is a geometric series with first term 1, common ratio 2 and n+1 terms. Hence the sum is: \(1+2+\cdots+2^n = \frac{1-2^{n+1}}{1-2} = 2^{n+1}-1.\) If this is greater than 500 (i.e. if the total amount of money in the account is more than $5 ) then \(2^{n+1}-1\ge 500\) , so \(2^{n+1}\ge 501\). The smallest power of 2 that is greater than 501 is \(2^9\). Thus the first time there is more than $5 in the bank account occurs after n=8 days. This is 8 days away from Sunday, so the day of the week is \(\boxed{\text{Monday}}\) .

azsun Jan 5, 2018