To solve this problem, we need to determine the amount in the account at month 6, given that the balances form a geometric sequence. Here's a step-by-step method to find the solution:
1. Identify the initial values and terms:
- Balance at month 1, [tex]\( A_1 = 1700 \)[/tex]
- Balance at month 2, [tex]\( A_2 = 2040 \)[/tex]
- Balance at month 3, [tex]\( A_3 = 2448 \)[/tex]
2. Determine the common ratio of the geometric sequence:
- The common ratio [tex]\( r \)[/tex] can be found by dividing the second term by the first term, and the third term by the second term:
[tex]\[
r = \frac{A_2}{A_1} = \frac{2040}{1700}
\][/tex]
[tex]\[
r = \frac{A_3}{A_2} = \frac{2448}{2040}
\][/tex]
- Both calculations give the common ratio:
[tex]\[
r = 1.2
\][/tex]
3. General formula for a geometric sequence:
- The [tex]\( n \)[/tex]-th term [tex]\( A_n \)[/tex] of a geometric sequence is given by:
[tex]\[
A_n = A_1 \cdot r^{n-1}
\][/tex]
4. Calculate the balance at month 6:
- We need to find [tex]\( A_6 \)[/tex]. Using the formula and substituting the known values:
[tex]\[
A_6 = 1700 \cdot (1.2)^{6-1}
\][/tex]
[tex]\[
A_6 = 1700 \cdot (1.2)^5
\][/tex]
5. Result:
- Carrying out the calculations, we find:
[tex]\[
A_6 \approx 4230.144
\][/tex]
Therefore, the amount in the account at month 6 is approximately \$4230.14.
