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Compare the investment below to an investment of the same principal at the same raye compund annually.
principal: $7000, annual interest: 5%, interest periods: 4, bumber of years: 15
After 15 years, the investmemt compund periodically will be worth __ more than the i vestment compund annually.

Respuesta :

To compare the two investments, we first need to calculate the future value of each investment.

For the investment compounded annually:
Using the compound interest formula:
\[ A = P(1 + \frac{r}{n})^{nt} \]
Where:
- \( P = $7000 \) (principal)
- \( r = 5\% = 0.05 \) (annual interest rate)
- \( n = 1 \) (compounded annually)
- \( t = 15 \) years

\[ A = 7000(1 + \frac{0.05}{1})^{1*15} \]
\[ A = 7000(1 + 0.05)^{15} \]
\[ A = 7000(1.05)^{15} \]
\[ A \approx 7000(2.078928) \]
\[ A \approx $14552.50 \]

For the investment compounded periodically:
Using the compound interest formula:
\[ A = P\left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- \( P = $7000 \) (principal)
- \( r = 5\% = 0.05 \) (annual interest rate)
- \( n = 4 \) (interest periods per year)
- \( t = 15 \) years

\[ A = 7000\left(1 + \frac{0.05}{4}\right)^{4*15} \]
\[ A = 7000\left(1 + 0.0125\right)^{60} \]
\[ A = 7000(1.0125)^{60} \]
\[ A \approx 7000(1.972647) \]
\[ A \approx $13808.53 \]

Now, let's find the difference in value between the two investments:
\[ Difference = A_{periodically} - A_{annually} \]
\[ Difference = $13808.53 - $14552.50 \]
\[ Difference \approx -$744.97 \]

Therefore, after 15 years, the investment compounded periodically will be worth approximately $744.97 less than the investment compounded annually.