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Find the accumulated amount \(A\) if the principal \(P\) is invested at the interest rate of \(r /\) year for \(t\) yr. $$ P=\$ 1000, r=7 \%, t=8, \text { compounded annually } $$

Short Answer

Expert verified
The accumulated amount after 8 years with the given principal amount, interest rate, and compounding frequency is approximately $1,716.69.
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Step by step solution

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Step 1: Identify the formula

To find the accumulated amount, we will use the compound interest formula: $$ A = P\left(1+\frac{r}{n}\right)^{nt} $$ where: - \(A\) is the accumulated amount (including principal and interest), - \(P\) is the principal amount (the initial amount invested), - \(r\) is the annual interest rate (in decimal form), - \(t\) is the time (in years), and - \(n\) is the number of times interest is compounded per year. Since the interest is compounded annually, \(n\) will be equal to 1.

Step 2: Convert interest rate to decimal

The given annual interest rate is 7%, so we need to convert it to decimal form by dividing by 100: \(r = \frac{7}{100} = 0.07\)

Step 3: Substitute the values into the formula

Now, we will substitute the given values into the formula: \(P = 1000\) \(n = 1\) \(r = 0.07\) \(t = 8\) $$ A = 1000\left(1+\frac{0.07}{1}\right)^{(1)(8)} $$

Step 4: Simplify and calculate the accumulated amount

Next, simplify the formula and calculate the accumulated amount: $$ A = 1000\left(1+0.07\right)^{8} $$ $$ A = 1000(1.07)^{8} $$ Using a calculator, we find that: $$ A \approx 1716.69 $$

Step 5: State the final answer

The accumulated amount after 8 years with the given principal amount, interest rate, and compounding frequency is approximately $1,716.69.

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