T 4–4 Deposits needed to accumulate a future sum

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In summary, Judi needs to make annual deposits of $1,256.25 at the end of each year for the next 5 years in order to accumulate $8,000 by the end of 5 years, assuming an annual interest rate of 7%. This can be calculated using the formula A=P(1+r/n)^(nt), where A is the future sum, P is the initial deposit, r is the interest rate, n is the number of compounding periods, and t is the number of years.
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karush
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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
 
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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 $D$:

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

FAQ: T 4–4 Deposits needed to accumulate a future sum

What is a T 4-4 deposit?

A T 4-4 deposit is a type of investment where a fixed amount of money is deposited at regular intervals, usually monthly, with the goal of accumulating a larger sum in the future.

How does a T 4-4 deposit work?

A T 4-4 deposit works by consistently investing a fixed amount of money over a set period of time. The deposits are usually made into a savings or investment account and earn interest, which helps to grow the overall sum.

What is the purpose of a T 4-4 deposit?

The main purpose of a T 4-4 deposit is to accumulate a larger sum in the future. This can be used for various purposes such as saving for a down payment on a house, funding a child's education, or building a retirement nest egg.

What factors should be considered when making T 4-4 deposits?

Some important factors to consider when making T 4-4 deposits include the interest rate, the length of the deposit period, and the frequency and amount of the deposits. It's also important to evaluate the risk associated with the investment and ensure it aligns with your financial goals.

Are there any risks associated with T 4-4 deposits?

Like any investment, T 4-4 deposits come with a certain level of risk. The value of the investment may fluctuate depending on market conditions and there is always a chance of losing some or all of the invested amount. It's important to carefully consider the risks and potential returns before making any investment decisions.

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