Chapter 12, Problem 6P

(a)

To determine

The recursive formula for the amount $U_n$ in Ursula’s CD at the end of the $n\text{th}$ month.

Answer to Problem 6P

The recursive formula for the amount $U_n$ in Ursula’s CD at the end of the $n\text{th}$ month is $U_n=\left(1.155\right)U_{n-1}$.

Explanation of Solution

Given:

Ursula deposit $5000 at the rate of 5% per year in a CD. She reinvests the CD amount at the rate of 5% and also 10% of the CD amount.

Calculation:

According to the given information the principle amount for first year is $5000 and the rate of interest r is 5% or 0.05. In the time of reinvest she add 10% of the CD amount in the CD amount. The initial value of investment is $5000, therefore the value of $U_0$ is 5000.

The total CD amount at the end of first year after addition of interest is calculated as follows,

$\begin{array}{c}U=5000+5000\left(0.05\right)\\ =U_0+U_0\left(0.05\right)\left(\because U_0=5000\right)\end{array}$

Ursula add 10% of CD amount at the time of reinvestment, so the total reinvest amount after 1 year is calculated as follows,

$\begin{array}{c}{U}_1=U_0+U_0\left(0.05\right)+\left(U_0+U_0\left(0.05\right)\right)\left(0.1\right)\\ =\left(1.05\right)U_0+U_0\left(1.05\right)\left(0.1\right)\\ =\left(1.05\right)U_0+\left(0.105\right)U_0\\ =\left(1.155\right)U_0\end{array}$

The new principle amount is $U_1=\left(1.155\right)U_0$.

The total CD amount at the end of second year after addition of interest is calculated as follows,

$U=U_1+U_1\left(0.05\right)$

Ursula add 10% of CD amount at the time of reinvestment, so the total reinvest amount after 2 year is calculated as follows,

$\begin{array}{c}{U}_2=U_1+U_1\left(0.05\right)+\left(U_1+U_1\left(0.05\right)\right)\left(0.1\right)\\ =\left(1.05\right)U_1+U_1\left(1.05\right)\left(0.1\right)\\ =\left(1.05\right)U_1+\left(0.105\right)U_1\\ =\left(1.155\right)U_1\end{array}$

The new principle amount is $\left(1.155\right)U_1$.

Similarly, the principle amount after n month is $U_n$ and Ursula add 10% of CD amount at the time of reinvestment, so the total reinvest amount after n year is calculated as follows,

$\begin{array}{c}{U}_n=U_{n-1}+U_{n-1}\left(0.05\right)+\left(U_{n-1}+U_{n-1}\left(0.05\right)\right)\left(0.1\right)\\ =\left(1.05\right)U_{n-1}+U_{n-1}\left(1.05\right)\left(0.1\right)\\ =\left(1.05\right)U_{n-1}+\left(0.105\right)U_{n-1}\\ =\left(1.155\right)U_{n-1}\end{array}$

Therefore, the recursive formula for the amount $U_n$ in Ursula’s CD at the end of the $n\text{th}$ month is $U_n=\left(1.155\right)U_{n-1}$.

(b)

To determine

The first five terms of the sequence $U_n$.

Answer to Problem 6P

The first five terms of the sequence $U_n$ are $U_0=5000$, $U_1=5775$, $U_2=6670.125$, $U_3=7703.99$ and $U_4=8898.11$.

Explanation of Solution

Given:

From part (a), the value of $U_n$ is given below,

$U_n=\left(1.155\right)U_{n-1}$ (1)

Calculation:

The initial amount is $5000. The value of first term $U_0$ is 5000.

Substitute 1 for n in equation (1), to find the value of $U_1$.

$\begin{array}{c}{U}_1=\left(1.155\right)U_{1-1}\\ =\left(1.155\right)U_0\\ =\left(1.155\right)\left(5000\right)\left(\because U_0=5000\right)\\ =5775\end{array}$

Substitute 2 for n in equation (1), to find the value of $U_2$.

$\begin{array}{c}{U}_2=\left(1.155\right)U_{2-1}\\ =\left(1.155\right)U_1\\ =\left(1.155\right)\left(5775\right)\left(\because U_1=5775\right)\\ =6670.125\end{array}$

Substitute 3 for n in equation (1), to find the value of $U_3$.

$\begin{array}{c}{U}_3=\left(1.155\right)U_{3-1}\\ =\left(1.155\right)U_2\\ =\left(1.155\right)\left(6670.125\right)\left(\because U_2=6670.125\right)\\ =7703.99\end{array}$

Substitute 4 for n in equation (1), to find the value of $U_4$.

$\begin{array}{c}{U}_4=\left(1.155\right)U_{4-1}\\ =\left(1.155\right)U_3\\ =\left(1.155\right)\left(7703.99\right)\left(\because U_3=7703.99\right)\\ =8898.11\end{array}$

Therefore, the first five terms of the sequence $U_n$ are $U_0=5000$, $U_1=5775$, $U_2=6670.125$, $U_3=7703.99$ and $U_4=8898.11$.

(c)

To determine

The formula for $U_n$ using the pattern of the terms in part (b).

Answer to Problem 6P

The formula for $U_n$ using the pattern of the terms in part (b) is $U_n=5000{\left(1.155\right)}^n$.

Explanation of Solution

Given:

The first five terms of the sequence $U_n$ are $U_0=5000$, $U_1=5775$, $U_2=6670.125$, $U_3=7703.99$ and $U_4=8898.11$.

Calculation:

The first term of the sequence $U_n$ is written below,

$A_0=5000$

The second term is written as,

$\begin{array}{c}{U}_1=\left(1.155\right)U_0\\ =\left(1.155\right)\left(5000\right)\left(\because U_0=5000\right)\end{array}$

The third term is written as,

$\begin{array}{c}{U}_2=\left(1.155\right)U_1\\ =\left(1.155\right)\left(1.155\right)\left(5000\right)\left(\because U_1=\left(1.155\right)\left(5000\right)\right)\\ ={\left(1.155\right)}^2\left(5000\right)\end{array}$

The fourth term is written as,

$\begin{array}{c}{U}_3=\left(1.155\right)U_2\\ =\left(1.155\right){\left(1.155\right)}^2\left(5000\right)\left(\because U_2={\left(1.155\right)}^2\left(5000\right)\right)\\ ={\left(1.155\right)}^3\left(5000\right)\end{array}$

The fifth term is written as,

$\begin{array}{c}{U}_4=\left(1.155\right)U_3\\ =\left(1.155\right){\left(1.155\right)}^3\left(5000\right)\left(\because U_3={\left(1.155\right)}^3\left(5000\right)\right)\\ ={\left(1.155\right)}^4\left(5000\right)\end{array}$

Similarly the general formula for $U_n$ is written as shown below,

$U_n=5000{\left(1.155\right)}^n$

Therefore, the formula for $U_n$ using the pattern of the terms in part (b) is $U_n=5000{\left(1.155\right)}^n$.

(d)

To determine

The total amount saved by Ursula after 10 years.

Answer to Problem 6P

The total amount saved by Ursula after 10 years is $21124.67.

Explanation of Solution

Given:

From part (c) the formula for $U_n$ is given below,

$U_n=5000{\left(1.155\right)}^n$ (2)

Where, $U_n$ is total saved amount after n years.

Calculation:

Substitute 10 for n in equation (2), to find the value of total saved amount after 10 years.

$\begin{array}{c}{U}_{10}=5000{\left(1.155\right)}^{10}\\ =5000\left(4.224933\right)\\ =21124.665\\ \approx 21124.67\end{array}$

Therefore, the total amount saved by Ursula after 10 years is $21124.67.