Chapter 8.1, Problem 53E

(a)

To determine

How much money you save after $100$ days.

Answer to Problem 53E

  $\$50.50$

Explanation of Solution

Given: You want to save $\$500$ for a school trip. You begin by saving a penny on the first day. You save an additional penny each day after that. For example you will save two penny on the second day, three penny on the third day and so on.

Here

We have to save $n$ pennies for $n$ days.

So the series can be written as

  $a_i=i$ Where $i=1,2,\mathrm{3...100}$

Thus

  $\begin{array}{l}{\displaystyle \sum _{i=1}^{100}i}=\frac{100(100+1)}{2}\\ =50(101)\\ =5050\end{array}$

Thus money saved after $100$ days is $5050\times \$0.01=\$50.50$

Hence, $\$50.50$

(b)

To determine

To Find: How many days it takes you to save $\$500$ .

Answer to Problem 53E

  $316days$

Explanation of Solution

Given: You want to save $\$500$ for a school trip. You begin by saving a penny on the first day. You save an additional penny each day after that. For example you will save two penny on the second day, three penny on the third day and so on.

To save $\$500$ we need to save $\$500\times 100=50,000$

Using the formula for sum of $n$ integers.

  $\begin{array}{l}\frac{d(d+1)}{2}=50,000\\ d(d+1)=100,000\\ d=315.73\\ d\approx 316days\end{array}$

Hence, $316days$