Chapter 12.3, Problem 93AYU

To determine

To find: The date and the amount.

Answer to Problem 93AYU

The date is 20^thDecember, 2011 and total amount received is $ 9,999.92.

Explanation of Solution

Given information:

First payment = $ 1000

Each day the payment get ${\frac{9}{10}}^{th}$ of the payment received the previous day.

Final payment = $1¢=\frac{1}{100}dollar$

Calculation:

Calculate the number of days (n)

Use formula: $a_n=a.r^{n-1}$

Consider the inequality:

  $\begin{array}{l}{a}_n<\frac{1}{100}dollar\\ $ 1000{(\frac{9}{10})}^{n-1}<\frac{1}{100}${\left(\frac{9}{10}\right)}^n<\frac{9}{{10}^6}\end{array}$

Therefore, the least value of n should be 111.

Therefore, after 111 days the amount will be less than 1¢.

Therefore, the date will be 1^st September + 111 days = 20^th December, 2011.

  $$ Use formula: $S_n=\frac{a(1-r^n)}{1-r}$

  $\begin{array}{c}{S}_{111}=\frac{1000\left[1-{\left(\frac{9}{10}\right)}^{111}\right]}{1-\frac{9}{10}}\\ =\frac{1000\left[1-{\left(\frac{9}{10}\right)}^{111}\right]}{\frac{1}{10}}\\ =9,999.92\end{array}$

Therefore, the amount received is $ 9,999.92.