Exercise 63 Page 432

Let's assume that we deposit $P$ dollars into an account. When the interest is compounded yearly with the interest rate equal to $r\%,$ the balance after $t$ years can be modeled with the formula for compound interest. In the first year we deposit $P=2000$ dollars in an account with interest rate $r=5\%=0.05.$ Let $y_{30}$ be the amount of money after $t=30$ years from that deposit. Let's find $y_{30}.$ Every year for $30$ years we deposit an additional $\$2000$ into the account. Let's look at how much money we will get from the deposit after the first year. We also deposit $P=2000$ dollars with the same interest rate $r=0.05.$ Let $y_{29}$ be the amount of money after $t=29$ years. Let's find $y_{29}.$ Therefore the total amount of our money after $30$ years is equal to the following partial sum of geometric series. Let's rearrange terms of the series. We will use the formula for finding partial sums of geometric series. For this series, the first term is $a_1=2000$ and the common ratio is $R=1.05.$ Since we are adding $31$ terms to represent the total $31$ years (including the first year you deposited), $n=31.$ Let's substitute the values and find the total amount of money in our account after $30$ years. Therefore, after $30$ years we will have about $\$141521.58$ in our account.