Here are two ways of investing $50,000 for 25 years. Lump-Sum Deposit Rate $50,000 4% compounded annually Periodic Deposit $2000 at the end of each year a. Rate 4% compounded annually Use this information and the formulas A = P(1 + r) and A = and b. below. ... Time 25 years Time 25 years P[(1+r)²-1] r to complete parts a. After 25 years, how much more will you have from the lump-sum investment than from the annuity? You will have approximately $ more from the lump-sum investment than from the annuity. (Round to the nearest dollar as needed.)
Here are two ways of investing $50,000 for 25 years. Lump-Sum Deposit Rate $50,000 4% compounded annually Periodic Deposit $2000 at the end of each year a. Rate 4% compounded annually Use this information and the formulas A = P(1 + r) and A = and b. below. ... Time 25 years Time 25 years P[(1+r)²-1] r to complete parts a. After 25 years, how much more will you have from the lump-sum investment than from the annuity? You will have approximately $ more from the lump-sum investment than from the annuity. (Round to the nearest dollar as needed.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Investment Analysis: Comparing Lump-Sum and Periodic Deposits**
This lesson explores two methods of investing $50,000 over a period of 25 years, both with an annual compounding interest rate of 4%. The options are outlined in the table below:
| Investment Type | Amount | Rate | Time |
|-------------------|------------------------|-------------------------|-----------|
| Lump-Sum Deposit | $50,000 | 4% compounded annually | 25 years |
| Periodic Deposit | $2,000 at the end of each year | 4% compounded annually | 25 years |
### Formulas to Know
1. **Lump-Sum Formula:**
\[
A = P(1 + r)^t
\]
where \( A \) is the amount of money accumulated after n years, including interest.
2. **Periodic Deposit (Annuity) Formula:**
\[
A = \frac{P[(1 + r)^t - 1]}{r}
\]
where \( A \) is the future value of the annuity and \( P \) is the periodic payment.
### Problem to Solve
**a. Comparing Future Values:**
After 25 years, calculate how much more the lump-sum investment yields compared to the annuity.
- **Lump-Sum Approach:** Deposit $50,000 once, compounded for 25 years.
- **Periodic Deposit Approach (Annuity):** Deposit $2,000 annually, compounded each year.
**Question: How much more will the lump-sum investment yield compared to the annuity?**
Fill in the blank with the approximate dollar difference, rounded to the nearest dollar.
**Additional Notes:**
The graph and table provide visual and numerical insights into how different investment strategies can impact the total returns over a 25-year period, emphasizing the time value of money and the power of compound interest.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89bbb75e-f0ee-48ed-9f1e-6a8f86b689c0%2F77a18f6d-5de8-4a4c-8770-0bae1c07912a%2F22bo9cm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Investment Analysis: Comparing Lump-Sum and Periodic Deposits**
This lesson explores two methods of investing $50,000 over a period of 25 years, both with an annual compounding interest rate of 4%. The options are outlined in the table below:
| Investment Type | Amount | Rate | Time |
|-------------------|------------------------|-------------------------|-----------|
| Lump-Sum Deposit | $50,000 | 4% compounded annually | 25 years |
| Periodic Deposit | $2,000 at the end of each year | 4% compounded annually | 25 years |
### Formulas to Know
1. **Lump-Sum Formula:**
\[
A = P(1 + r)^t
\]
where \( A \) is the amount of money accumulated after n years, including interest.
2. **Periodic Deposit (Annuity) Formula:**
\[
A = \frac{P[(1 + r)^t - 1]}{r}
\]
where \( A \) is the future value of the annuity and \( P \) is the periodic payment.
### Problem to Solve
**a. Comparing Future Values:**
After 25 years, calculate how much more the lump-sum investment yields compared to the annuity.
- **Lump-Sum Approach:** Deposit $50,000 once, compounded for 25 years.
- **Periodic Deposit Approach (Annuity):** Deposit $2,000 annually, compounded each year.
**Question: How much more will the lump-sum investment yield compared to the annuity?**
Fill in the blank with the approximate dollar difference, rounded to the nearest dollar.
**Additional Notes:**
The graph and table provide visual and numerical insights into how different investment strategies can impact the total returns over a 25-year period, emphasizing the time value of money and the power of compound interest.
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