To calculate how much money you will have after 10 years when investing $1,000 in an account with an annual interest rate of 5%, compounded yearly, you can use the formula for compound interest, which follows the same structure as a geometric sequence.

The formula for compound interest is:

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

Where:

• $A$ is the amount of money after time $t$.
• $P$ is the principal (initial investment), which is $1,000 in this case.
• $r$ is the annual interest rate (5% or 0.05).
• $n$ is the number of times the interest is compounded per year (since it is compounded yearly, $n = 1$).
• $t$ is the number of years (in this case, $t = 10$).

Plugging in the values:

$A = 1000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 10}$

$A = 1000 \times (1.05)^{10}$

Now, we calculate $(1.05)^{10}$:

$A = 1000 \times 1.62889 \approx 1628.89$

After 10 years, the investment will grow to approximately $1,628.89.

Would you like to see more details, or have any questions?

Here are 5 related questions to explore:

1. What happens if the interest is compounded quarterly instead of yearly?
2. How would the amount change if the interest rate was 3% instead of 5%?
3. Can we derive a general formula for different time periods and rates?
4. How does continuous compounding compare to yearly compounding?
5. What is the difference between simple interest and compound interest?

Tip: The more frequently interest is compounded, the more your investment will grow over time due to the effect of "interest on interest."