Chapter 9.5, Problem 16PPSE

Solution

To find if the family will have enough money for the trip or not.

The family will not have enough money. They will need $\$134.76$ more.

Given information:

A family starts saving money in March with $125 and increases the amount by 20% every month, and they need to have $1375 at the end of August for the trip.

Calculation:

The amount saved each month will be:

  $\begin{array}{l}{a}_1=125\\ a_2=1.2a_1=1.2\cdot 125\\ a_3=1.2a_2=1.2\cdot 1.2\cdot 125=125{\left(1.2\right)}^2\\ a_4=1.2a_3=1.2\cdot 125{\left(1.2\right)}^2=125{\left(1.2\right)}^3\\ a_5=1.2a_4=1.2\cdot 125{\left(1.2\right)}^3=125{\left(1.2\right)}^4\\ a_6=1.2a_5=1.2\cdot 125{\left(1.2\right)}^4=125{\left(1.2\right)}^5\end{array}$

Therefore, the total amount saved from March to August can be given by the series,

  $125+125\left(1.2\right)+125{\left(1.2\right)}^2+125{\left(1.2\right)}^3+125{\left(1.2\right)}^4+125{\left(1.2\right)}^5$

This is a geometric series with $a_1=125$ and $r=1.2$ .

Now, find the sum of the series.

  $\begin{array}{c}{S}_6=\frac{125\left(1-{\left(1.2\right)}^6\right)}{1-1.2}\\ =\frac{125\left(-1.985984\right)}{-0.2}\\ =\text{1241}\text{.24}\end{array}$

So, the amount saved by the family at end of August will be $1241.24. That means, the family will not have enough money for the trip. They need more $\$1375-\$1241.24=\$134.76$ .