If $$ \$ 500 $$ are deposited into an account with a $$ 4.5 \% $$ interest rate, compounded annually, what is the balance after 6 years? $$ F=\$[\text { ? ] } $$ $$ F=P\left(1+\frac{r}{n}\right)^{\text {nt }} $$

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UpStudy AI · Accepted Answer

To find the balance after 6 years, we can use the formula for compound interest:

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

Given:
- Principal amount ($$ P $$) = $500
- Interest rate ($$ r $$) = 4.5% = 0.045
- Compounding frequency ($$ n $$) = 1 (compounded annually)
- Time ($$ t $$) = 6 years

Substitute the given values into the formula:

$$ F = 500\left(1 + \frac{0.045}{1}\right)^{1 \times 6} $$

Now, we can calculate the balance after 6 years.
Calculate the value by following steps:
- step0: Calculate:
  $$500\left(1+0.045\right)^{6}$$
- step1: Add the numbers:
  $$500\times 1.045^{6}$$
- step2: Convert the expressions:
  $$500\left(\frac{209}{200}\right)^{6}$$
- step3: Simplify:
  $$500\times \frac{209^{6}}{200^{6}}$$
- step4: Rewrite the expression:
  $$100\times 5\times \frac{209^{6}}{200^{6}}$$
- step5: Rewrite the expression:
  $$100\times 5\times \frac{209^{6}}{100^{6}\times 2^{6}}$$
- step6: Reduce the numbers:
  $$5\times \frac{209^{6}}{100^{5}\times 2^{6}}$$
- step7: Rewrite the expression:
  $$5\times \frac{209^{6}}{25^{5}\times 4^{5}\times 2^{6}}$$
- step8: Rewrite the expression:
  $$5\times \frac{209^{6}}{5^{10}\times 4^{5}\times 2^{6}}$$
- step9: Reduce the numbers:
  $$1\times \frac{209^{6}}{5^{9}\times 4^{5}\times 2^{6}}$$
- step10: Multiply the fractions:
  $$\frac{209^{6}}{2^{16}\times 5^{9}}$$
- step11: Evaluate the power:
  $$\frac{209^{6}}{65536\times 5^{9}}$$
The balance after 6 years is approximately $651.13.
The balance after 6 years is approximately $651.13.

If are deposited into an account with
a interest rate, compounded
annually, what is the balance after 6 years?

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If are deposited into an account with a interest rate, compounded annually, what is the balance after 6 years?