Question

Evaluate the following expression for Pequals2000​, requals5​%, kequals4​, and nequals16.
Upper P left parenthesis 1 plus StartFraction r Over k EndFraction right parenthesis Superscript kn
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Part 1
Upper P left parenthesis 1 plus StartFraction r Over k EndFraction right parenthesis Superscript kn
equals
  
enter your response here
​(Do not round until the final answer. Then round to two decimal places as​ needed.)

Answer 1

To evaluate the expression

\[
P \left(1 + \frac{r}{k}\right)^{kn}
\]

we need to substitute the given values: \(P = 2000\), \(r = 5\%\) (or \(0.05\) in decimal), \(k = 4\), and \(n = 16\).

1. First, calculate \(\frac{r}{k}\):

\[
\frac{r}{k} = \frac{0.05}{4} = 0.0125
\]

2. Next, calculate \(kn\):

\[
kn = 4 \times 16 = 64
\]

3. Now substitute these values into the expression:

\[
P \left(1 + \frac{r}{k}\right)^{kn} = 2000 \left(1 + 0.0125\right)^{64}
\]

4. Calculate \(1 + 0.0125\):

\[
1 + 0.0125 = 1.0125
\]

5. Now raise \(1.0125\) to the power of \(64\):

\[
1.0125^{64}
\]

Using a calculator:

\[
1.0125^{64} \approx 1.898
\]

6. Now multiply this result by \(P = 2000\):

\[
2000 \times 1.898 \approx 3796
\]

Finally, round to two decimal places if needed:

\[
\text{Final Result} = \boxed{3796.00}
\]