Certainly! Let's determine the [tex]\( n \)[/tex]th term of the given sequence:
Given the sequence:
[tex]\[ \_ , 6, 4, 6, 4, 6, \ldots \][/tex]
### Step-by-Step Solution:
1. Analyze the Sequence Pattern:
- The given sequence appears to alternate between two numbers: 6 and 4.
- Observing the terms: the sequence alternates starting with 6, followed by 4.
2. Determine the Position Dependence:
- The first term (which is missing) is unknown, but the given sequence starting from the second position follows a pattern:
- 2nd term is 6
- 3rd term is 4
- 4th term is 6
- 5th term is 4
- 6th term is 6
- and so on...
- We can see the pattern:
- If the position is odd (1st, 3rd, 5th, ...), the term is 4.
- If the position is even (2nd, 4th, 6th, ...), the term is 6.
3. Find the [tex]\( n \)[/tex]th Term Rule:
- If [tex]\( n \)[/tex] is odd, the [tex]\( n \)[/tex]th term is 4.
- If [tex]\( n \)[/tex] is even, the [tex]\( n \)[/tex]th term is 6.
4. Apply the Rule to Any Position [tex]\( n \)[/tex]:
- Check if [tex]\( n \)[/tex] is odd or even.
- If [tex]\( n \)[/tex] is odd ([tex]\( n \% 2 == 1 \)[/tex]), then [tex]\( a_n = 4 \)[/tex].
- If [tex]\( n \)[/tex] is even ([tex]\( n \% 2 == 0 \)[/tex]), then [tex]\( a_n = 6 \)[/tex].
### Conclusion:
To find the [tex]\( n \)[/tex]th term of the sequence:
[tex]\[ a_n = \begin{cases}
6 & \text{if } n \text{ is odd} \\
4 & \text{if } n \text{ is even}
\end{cases} \][/tex]
Using this rule, you can determine the [tex]\( n \)[/tex]th term for any position [tex]\( n \)[/tex] in the sequence.
