Certainly! Let's address the given sequence and solve the problem step-by-step.
### Part (a): Determine the Geometric Sequence Rule
A geometric sequence is one in which each term after the first is found by multiplying the previous term by a constant ratio, known as the common ratio (r).
Given the terms of the sequence: 900, 720, 576, 460.8, ...
To find the common ratio (r), we divide the second term by the first term:
[tex]\[ r = \frac{720}{900} = \frac{4}{5} = 0.8 \][/tex]
To verify that this ratio is consistent for the other terms:
[tex]\[ \frac{576}{720} = \frac{4}{5} = 0.8 \][/tex]
[tex]\[ \frac{460.8}{576} = \frac{4}{5} = 0.8 \][/tex]
The common ratio (r) is indeed 0.8.
Therefore, the geometric sequence rule can be written as the [tex]\( n \)[/tex]-th term of the sequence, [tex]\( a_n \)[/tex], given by:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
where [tex]\( a_1 = 900 \)[/tex] and [tex]\( r = 0.8 \)[/tex].
So, the sequence rule is:
[tex]\[ a_n = 900 \cdot 0.8^{(n-1)} \][/tex]
### Part (b): Find the 8th Term of the Sequence
Now, we need to find the 8th term of the sequence:
[tex]\[ a_8 = 900 \cdot 0.8^{(8-1)} \][/tex]
[tex]\[ a_8 = 900 \cdot 0.8^7 \][/tex]
Calculating [tex]\( 0.8^7 \)[/tex]:
[tex]\[ 0.8^7 = 0.2097152 \][/tex]
Now, multiplying this by 900:
[tex]\[ a_8 = 900 \cdot 0.2097152 \][/tex]
[tex]\[ a_8 = 188.74368 \][/tex]
Rounding to two decimal places:
[tex]\[ a_8 \approx 188.74 \][/tex]
So, the 8th term of the sequence is:
[tex]\[ a_8 = 188.74 \][/tex]
Summarizing our findings:
- The geometric sequence rule is [tex]\( a_n = 900 \cdot 0.8^{(n-1)} \)[/tex].
- The 8th term of the sequence is [tex]\( 188.74 \)[/tex] when rounded to two decimal places.
