To determine the first three terms of the sequence whose [tex]\( n \)[/tex]th term is given by the formula [tex]\( A_n = \left(\frac{3}{4}\right)^n \)[/tex], we will substitute [tex]\( n = 1 \)[/tex], [tex]\( n = 2 \)[/tex], and [tex]\( n = 3 \)[/tex] into the formula.
### Step-by-Step Solution:
1. First Term ([tex]\( A_1 \)[/tex]):
- Substitute [tex]\( n = 1 \)[/tex] into the formula [tex]\( A_n = \left(\frac{3}{4}\right)^n \)[/tex]:
[tex]\[
A_1 = \left(\frac{3}{4}\right)^1 = \frac{3}{4}
\][/tex]
2. Second Term ([tex]\( A_2 \)[/tex]):
- Substitute [tex]\( n = 2 \)[/tex] into the formula:
[tex]\[
A_2 = \left(\frac{3}{4}\right)^2 = \left(\frac{3}{4} \times \frac{3}{4}\right) = \frac{9}{16}
\][/tex]
- The decimal equivalent is:
[tex]\[
\frac{9}{16} = 0.5625
\][/tex]
3. Third Term ([tex]\( A_3 \)[/tex]):
- Substitute [tex]\( n = 3 \)[/tex] into the formula:
[tex]\[
A_3 = \left(\frac{3}{4}\right)^3 = \left(\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}\right) = \frac{27}{64}
\][/tex]
- The decimal equivalent is:
[tex]\[
\frac{27}{64} = 0.421875
\][/tex]
### Summary:
The first three terms of the sequence given by [tex]\( A_n = \left(\frac{3}{4}\right)^n \)[/tex] are:
[tex]\[
A_1 = 0.75
\][/tex]
[tex]\[
A_2 = 0.5625
\][/tex]
[tex]\[
A_3 = 0.421875
\][/tex]
Therefore, the first three terms of the sequence are [tex]\( 0.75 \)[/tex], [tex]\( 0.5625 \)[/tex], and [tex]\( 0.421875 \)[/tex].
