Certainly! Let's break this problem down step-by-step.
### 1. Understanding the Problem
We need to determine the amount of money accumulated after 6 years if [tex]$700 is invested at an annual interest rate of 2.25%, compounded quarterly.
### 2. Compounded Interest Formula
The formula for compounded interest is given by:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
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 for.
### 3. Substituting the Given Values
From the problem, we have:
- \( P = 700 \) (Principal)
- \( r = 2.25\% = 0.0225 \) (Annual interest rate as a decimal)
- \( n = 4 \) (Interest is compounded quarterly, hence 4 times per year)
- \( t = 6 \) (Number of years)
Substituting these values into the formula:
\[ A = 700 \left(1 + \frac{0.0225}{4}\right)^{4 \times 6} \]
### 4. Simplifying the Formula
Let's simplify the expression inside the parentheses first:
\[ \frac{0.0225}{4} = 0.005625 \]
Now the formula becomes:
\[ A = 700 \left(1 + 0.005625 \right)^{24} \]
\[ A = 700 \left(1.005625 \right)^{24} \]
### 5. Computing the Final Amount
Raising \( 1.005625 \) to the power of 24 and then multiplying by 700:
\[ A \approx 700 \times 1.14396106555 \]
\[ A \approx 800.8727458864043 \]
### 6. Conclusion
After 6 years, the amount accumulated with a $[/tex]700 principal invested at an annual interest rate of 2.25%, compounded quarterly, is approximately $800.87.
### Explicit Geometric Sequence Formula
For a geometric sequence with common ratio [tex]\( r = 1.005625 \)[/tex] and initial term [tex]\( a = 700 \)[/tex], the explicit formula for the nth term (which represents the value at each quarter) can be written as:
[tex]\[ A_n = 700 \times (1.005625)^n \][/tex]
where [tex]\( n \)[/tex] is the total number of compounding periods. For 6 years compounded quarterly, [tex]\( n = 4 \times 6 = 24 \)[/tex].
Therefore, the explicit formula for the geometric sequence representing the value of the investment at each quarter is:
[tex]\[ A_n = 700 \times (1.005625)^n \][/tex]
