Chapter 11.3, Problem 75HP

To determine

Find the amount of money in her account originally.

Answer to Problem 75HP

She had $32,000 in the savings account originally.

Explanation of Solution

Given:

Danette has a savings account. She withdraws half of the contents every year. After 4 years, she has $2000 left. How much did she have in the savings account originally?

Concept Used:

The amounts in the account form a geometric sequence with $a_5=2000\text{and}r=0.5$

Use the formula for the nth term of a geometric sequence: $a_n=a_1·r^{n-1}$

Calculation:

The amounts in the account form a geometric sequence with $a_5=2000\text{and}r=0.5$

Use the formula for the nth term of a geometric sequence: $a_n=a_1·r^{n-1}$

  $\begin{array}{l}{a}_5=2000;r=0.5;\text{find}{a}_1\\ a_n=a_1·r^{n-1}\\ a_5=a_1·r^{5-1}=a_1·r^4\\ $ 2000=a\cdot {(0.5)}^4=0.0625\cdot a_1$\frac{2000}{0.0625}=\frac{0.0625·a_1}{0.0625}\\ a_1=32000\end{array}$

Thus, she had $32,000 in the savings account originally.