Certainly! Let's determine the values for the 5th, 6th, 7th, and 8th terms in the sequence based on the details provided.
### Step-by-Step Solution:
1. Identify the Pattern:
We are given the first four terms of the sequence:
- 1st term: 320
- 2nd term: 160
- 3rd term: 80
- 4th term: 40
Observing these values, it is clear that each term is half of the previous term. This tells us that the sequence is a geometric progression where the common ratio ([tex]\( r \)[/tex]) is [tex]\( \frac{1}{2} \)[/tex].
2. Verify the Common Ratio:
To confirm the pattern, let's check the ratio between consecutive terms:
- Ratio between 1st and 2nd term: [tex]\( \frac{160}{320} = \frac{1}{2} \)[/tex]
- Ratio between 2nd and 3rd term: [tex]\( \frac{80}{160} = \frac{1}{2} \)[/tex]
- Ratio between 3rd and 4th term: [tex]\( \frac{40}{80} = \frac{1}{2} \)[/tex]
This confirms that the common ratio [tex]\( r \)[/tex] is indeed [tex]\( \frac{1}{2} \)[/tex].
3. Calculate the 5th Term:
Using the common ratio, the 5th term is:
[tex]\[
\text{5th term} = \text{4th term} \times \frac{1}{2} = 40 \times \frac{1}{2} = 20
\][/tex]
4. Calculate the 6th Term:
Using the same ratio, the 6th term is:
[tex]\[
\text{6th term} = \text{5th term} \times \frac{1}{2} = 20 \times \frac{1}{2} = 10
\][/tex]
5. Calculate the 7th Term:
Continuing this pattern, the 7th term is:
[tex]\[
\text{7th term} = \text{6th term} \times \frac{1}{2} = 10 \times \frac{1}{2} = 5
\][/tex]
6. Calculate the 8th Term:
Lastly, the 8th term is:
[tex]\[
\text{8th term} = \text{7th term} \times \frac{1}{2} = 5 \times \frac{1}{2} = 2.5
\][/tex]
### Final Values:
- 5th term: 20
- 6th term: 10
- 7th term: 5
- 8th term: 2.5
Therefore, the sequence looks like this:
[tex]\[
320, 160, 80, 40, 20, 10, 5, 2.5
\][/tex]
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline
\text{Term} & 1\text{st} & 2\text{nd} & 3\text{rd} & 4\text{th} & 5\text{th} & 6\text{th} & 7\text{th} & 8\text{th} \\
\hline
\text{Value} & 320 & 160 & 80 & 40 & 20 & 10 & 5 & 2.5 \\
\hline
\end{array}
\][/tex]
These calculations give us a complete understanding of the values in the series from the 1st to the 8th term.
