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.