Here are two ways of investing ​$5 comma 000 for 10 years.
  ​ Lump-Sum Deposit
   Rate
   Time
   ​$5 comma 000
   10​% compounded annually
   10 years
   Periodic Deposit
   Rate
   Time
   ​$500 at the end of
   each year
   10​% compounded annually
   10 years
Use this information and the formulas AequalsUpper P left parenthesis 1 plus r right parenthesis Superscript t and AequalsStartFraction Upper P left bracket left parenthesis 1 plus r right parenthesis Superscript t Baseline minus 1 right bracket Over r EndFraction
 to complete parts a. and b. below.
Question content area bottom
Part 1
a.
After 10 ​years, how much more will you have from the​ lump-sum investment than from the​ annuity?
You will have approximately ​$
  
enter your response here more from the​ lump-sum investment than from the annuity.
​(Round to the nearest dollar as​ needed.) 


We are given two types of investments: a **lump-sum deposit** and a **periodic deposit** (annuity), both compounding annually at a rate of 10% for 10 years. Let's calculate the final amount for both options.

### Formulas:
1. **Lump-Sum Investment**:
   \[
   A = P(1 + r)^t
   \]
   Where:
   - $ A $ is the amount after $ t $ years,
   - $ P $ is the initial principal,
   - $ r $ is the annual interest rate,
   - $ t $ is the time in years.

2. **Annuity (Periodic Deposit)**:
   \[
   A = \frac{P \left( (1 + r)^t - 1 \right)}{r}
   \]
   Where:
   - $ P $ is the periodic deposit,
   - $ r $ is the annual interest rate,
   - $ t $ is the time in years.

### Step 1: Calculate the Lump-Sum Investment Amount

Using the formula for the lump sum:
- $ P = 5000 $ (the initial deposit),
- $ r = 0.10 $ (10% interest rate),
- $ t = 10 $ years.

\[
A_{\text{lump}} = 5000(1 + 0.10)^{10}
\]

### Step 2: Calculate the Periodic Deposit (Annuity) Amount

Using the annuity formula:
- $ P = 500 $ (the amount deposited each year),
- $ r = 0.10 $ (10% interest rate),
- $ t = 10 $ years.

\[
A_{\text{annuity}} = \frac{500 \left( (1 + 0.10)^{10} - 1 \right)}{0.10}
\]

Let's now calculate both values.After 10 years:

- The lump-sum investment will grow to approximately **$12,969**.
- The periodic deposit (annuity) will grow to approximately **$7,969**.

The difference between the lump-sum investment and the annuity is approximately **$5,000**. Therefore, you will have **$5,000 more** from the lump-sum investment than from the annuity.

Would you like further details or clarification on any step?

Here are 5 related questions to consider:
1. What if the interest rate were compounded semi-annually instead of annually?
2. How would the result change if the time period was extended to 20 years?
3. What happens if the periodic deposit amount increases every year?
4. How is the total amount affected if a lower interest rate is applied?
5. How do inflation and purchasing power impact the value of these investments over time?

**Tip**: Annuities are useful for long-term saving strategies, but lump-sum investments grow faster due to early compounding.

Here are two ways of investing ​$5,000 for 10 years. Lump-Sum Deposit: $5,000, 10% compounded annually, 10 years. Periodic Deposit: $500 at the end of each year, 10% compounded annually, 10 years. Use the formulas A = P(1 + r)^t and A = [P((1 + r)^t - 1)] / r to complete parts a and b. After 10 years, how much more will you have from the lump-sum investment than from the annuity?

Explore how a $5,000 lump-sum deposit and $500 annual periodic deposits grow over 10 years at a 10% annual interest rate. Learn to calculate both scenarios using compound interest formulas and determine which investment yields more. Suitable for students in Grades 10-12.

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