To calculate the amount obtained after investing $1000 for 5 years at an interest rate of 5%, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the number of years the money is invested or borrowed.
Now, let's calculate the amount for both compounding methods:
(i) Compounded Yearly
Here, \( P = 1000 \), \( r = 0.05 \), \( n = 1 \) (compounded once a year), and \( t = 5 \).
\[ A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \times 5} \] \[ A = 1000 \left(1 + 0.05\right)^{5} \] \[ A = 1000 \left(1.05\right)^{5} \] \[ A = 1000 \times 1.2762815625 \quad \text{(calculated using a calculator)} \] \[ A \approx 1276.28 \]
(ii) Compounded Semi-Annually
For semi-annual compounding, \( P = 1000 \), \( r = 0.05 \), \( n = 2 \) (compounded twice a year), and \( t = 5 \).
\[ A = 1000 \left(1 + \frac{0.05}{2}\right)^{2 \times 5} \] \[ A = 1000 \left(1 + 0.025\right)^{10} \] \[ A = 1000 \left(1.025\right)^{10} \] \[ A = 1000 \times 1.280084528 \quad \text{(calculated using a calculator)} \] \[ A \approx 1280.08 \]
Summary of Results:
- Compounded Yearly: approximately $1276.28
- Compounded Semi-Annually: approximately $1280.08
So, the final amounts are approximately $1276.28 for yearly compounding and $1280.08 for semi-annual compounding.