To determine the [tex]\( n \)[/tex]th term of the given sequence [tex]\( 4, 6, 4, 6, 4, 6, \ldots \)[/tex], we can observe the pattern repeating periodically. Here is a step-by-step solution:
1. Identify the pattern:
- The sequence alternates between 4 and 6.
2. Formulate the general rule:
- Let's examine the positions of the terms:
- [tex]\( a_1 = 4 \)[/tex]
- [tex]\( a_2 = 6 \)[/tex]
- [tex]\( a_3 = 4 \)[/tex]
- [tex]\( a_4 = 6 \)[/tex]
- Notice that:
- For odd indices (1, 3, 5, ...), the term is 4.
- For even indices (2, 4, 6, ...), the term is 6.
3. Determine the [tex]\( n \)[/tex]th term:
- If the index [tex]\( n \)[/tex] is odd, the term at [tex]\( a_n \)[/tex] is 4.
- If the index [tex]\( n \)[/tex] is even, the term at [tex]\( a_n \)[/tex] is 6.
4. Write the final result:
- If [tex]\( n \)[/tex] is odd:
[tex]\[
a_n = 4
\][/tex]
- If [tex]\( n \)[/tex] is even:
[tex]\[
a_n = 6
\][/tex]
Thus, the [tex]\( n \)[/tex]th term [tex]\( a_n \)[/tex] of the sequence [tex]\( 4, 6, 4, 6, 4, 6, \ldots \)[/tex] is 4 if [tex]\( n \)[/tex] is odd, and 6 if [tex]\( n \)[/tex] is even.
