T 4–4 Deposits needed to accumulate a future sum

[karush]

T 4–4 Deposits needed to accumulate a future sum Judi wishes to accumulate \$8,000 by the end of 5 years by making equal annual end-of-year deposits over the next 5 years. If Judi can earn 7% on her investments, how much must she deposit at the end of each year to meet this goal?

$$\displaystyle A=P\left(1+\frac{r}{n}\right)^{nt}$$

ok not sure how plug this in

this complicated by the deposit made at the end of each year

 

[MarkFL]

Let's let $D$ be the amount of the end of year deposits, and $i$ be the annual interest rate. So, the amount of the account at the end of year $n$ can be given by the difference equation:

\(\displaystyle A_n-(1+i)A_{n-1}=D\) where $n\in\mathbb{N}$

The homogeneous solution is given by:

\(\displaystyle h_n=k_1(1+i)^n\)

And the particular solution is:

\(\displaystyle p_n=k_2\)

Plugging this into our difference equation, we find:

\(\displaystyle k_2-(1+i)k_2=D\implies k_2=-\frac{D}{i}\)

And so the closed form for $A_n$ is given by:

\(\displaystyle A_n=k_1(1+i)^n-\frac{D}{i}\)

Since:

\(\displaystyle A_1=D\)

We find:

\(\displaystyle k_1=\frac{D}{i}\)

And so the closed-form for $A_n$ is

\(\displaystyle A_n=\frac{D}{i}\left((1+i)^n-1\right)\)

Solve for $D$:

\(\displaystyle D=\frac{A_ni}{(1+i)^n-1}\)

 

[karush]

good grief,,