Chapter 9.1, Problem 70PPSE

Solution

a.

To write a recursive definition and an explicit formula for the amount of money at the end of any week.

The recursive definition is $a_n=a_{n-1}+5$ , where $a_1=25$ , and the explicit formula is $a_n=5n+20$ .

Given information:

Bank offers to start a saving account depositing $25 and depositing $5 a week starting the next week.

Calculation:

The initial amount is $25. So, $a_1=25$ .

The amount at the end of second, third and fourth week will be:

  $\begin{array}{l}{a}_2=a_1+5\\ a_3=a_2+5\\ a_4=a_5+5\end{array}$

So, the recursive definition can be written as $a_n=a_{n-1}+5$ , where $a_1=25$ .

The values of the amounts at the end of the first four week will be:

  $\begin{array}{l}{a}_1=25\\ a_2=a_1+5=25+5=30\\ a_3=a_2+5=30+5=35\\ a_4=a_5+5=35+5=40\end{array}$

This represents an arithmetic sequence with $a_1=25$ and $d=5$ . So, the nth term of the sequence will be:

  $\begin{array}{c}{a}_n=a_1+\left(n-1\right)d\\ =25+\left(n-1\right)5\\ =5n+20\end{array}$

b.

To find the amount in the bank after 4 weeks.

The amount in the bank at the end of the 4^th week will be $40.

Given information:

Bank offers to start a saving account depositing $25 and depositing $5 a week starting the next week.

Calculation:

From part (a), the amount at the end of any week can be obtained using the formula, $a_n=5n+20$ .

Substitute $n=4$ in $a_n=5n+20$ .

  $\begin{array}{c}{a}_n=5n+20\\ =5\cdot 4+20\\ =20+20\\ =40\end{array}$

Therefore, the amount in the bank at the end of the 4^th week will be $40.

c.

To write the recursive formula for the amount of money after every interest payment.

The recursive formula for the amount after every interest paid is $a_{4n}=1.005a_{4n-1}+1.025$ , where $a_3=35$ .

Given information:

Bank offers to start a saving account depositing $25 and depositing $5 a week starting the next week.

Calculation:

The recursive formula from part (a) is $a_n=a_{n-1}+5$ where $a_1=25$ .

The recursive formula for the amount at the end of every four week can be given as $a_{4n}=a_{4n-1}+5$ , where $a_3=35$ .

After the interest payment the amount will be:

  $\begin{array}{c}{a}_{4n}=\left(a_{4n-1}+5\right)+0.005\left(a_{4n-1}+5\right)\\ =1.005\left(a_{4n-1}+5\right)\\ =1.005a_{4n-1}+1.025\end{array}$

Therefore, the recursive formula for the amount after every interest paid is $a_{4n}=1.005a_{4n-1}+1.025$ , where $a_3=35$ .

d.

To find the bank’s annual interest rate.

The annual interest rate of the bank is $6\%$ .

Given information:

Bank offers to start a saving account depositing $25 and depositing $5 a week starting the next week.

Calculation:

Since the amount at every four week is multiplied by $0.005$ , the monthly interest is 0.5%.

Multiply this by 12 to get the annual interest rate of the bank.

So, the annual interest rate of the bank is $12\times 0.5\%=6\%$ .