Find the final amount of money in an account if $$ \$ 5,200 $$ is deposited at $$ 5.5 \% $$ interest compounded semi-annually and the money is left for 6 years.

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To find the final amount of money in an account with compound interest, 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.

Given:
- $$ P = 5200 $$
- $$ r = 5.5\% = 0.055 $$
- $$ n = 2 $$ (since the interest is compounded semi-annually)
- $$ t = 6 $$

Now, we can substitute these values into the formula:

$$
A = 5200 \left(1 + \frac{0.055}{2}\right)^{2 \times 6}
$$

Let's calculate the expression step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.055}{2} = 0.0275
   $$

2. Calculate $$ nt $$:
   $$
   2 \times 6 = 12
   $$

3. Now substitute these values into the formula:
   $$
   A = 5200 \left(1 + 0.0275\right)^{12}
   $$

4. Calculate $$ 1 + 0.0275 $$:
   $$
   1 + 0.0275 = 1.0275
   $$

5. Now raise $$ 1.0275 $$ to the power of $$ 12 $$:
   $$
   (1.0275)^{12}
   $$

6. Finally, multiply by $$ 5200 $$.

Let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$1.0275^{12}$$
- step1: Convert the expressions:
  $$\left(\frac{411}{400}\right)^{12}$$
- step2: Use the properties of exponents:
  $$\frac{411^{12}}{400^{12}}$$
Calculate or simplify the expression $$ 5200 * (1.0275^12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$5200\times 1.0275^{12}$$
- step1: Convert the expressions:
  $$5200\left(\frac{411}{400}\right)^{12}$$
- step2: Simplify:
  $$5200\times \frac{411^{12}}{400^{12}}$$
- step3: Rewrite the expression:
  $$400\times 13\times \frac{411^{12}}{400^{12}}$$
- step4: Reduce the numbers:
  $$13\times \frac{411^{12}}{400^{11}}$$
- step5: Multiply:
  $$\frac{13\times 411^{12}}{400^{11}}$$
After performing the calculations, we find:

1. $$ (1.0275)^{12} \approx 1.384784 $$

2. Now, we can calculate the final amount $$ A $$:
   $$
   A \approx 5200 \times 1.384784 \approx 7200.88
   $$

Thus, the final amount of money in the account after 6 years is approximately **\$7,200.88**.
The final amount in the account after 6 years is approximately \$7,200.88.

Find the final amount of money in an account if \( \$ 5,200 \) is deposited at \( 5.5 \% \) interest compounded semi-annually and the money is left for 6 years.

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