Let's carefully check each of the statements concerning the sequence [tex]\(-3, 5, -7, 9, -11, \ldots\)[/tex]:
1. The sequence has 5 terms.
The sequence given extends with the pattern shown. Even though we see 5 terms, the sequence itself doesn't necessarily stop there. A sequence can be infinite unless specified otherwise. Therefore, this statement is misleading because it implies the sequence has a finite number of terms.
2. The 4th term of the sequence is 9.
Let's check the 4th term in the given sequence:
[tex]\[
\begin{array}{cc}
\text{1st term} & -3 \\
\text{2nd term} & 5 \\
\text{3rd term} & -7 \\
\text{4th term} & 9 \\
\end{array}
\][/tex]
Indeed, the 4th term of the sequence is 9. This statement is correct.
3. [tex]\( f(5) = 2 \)[/tex]
In the context of sequences, [tex]\( f(n) \)[/tex] generally represents the [tex]\(n\)[/tex]-th term of the sequence. Looking at the 5th term:
[tex]\[
\text{5th term} = -11
\][/tex]
Therefore, [tex]\( f(5) = -11 \)[/tex], not 2. This statement is false.
4. The domain of the sequence is all natural numbers.
The domain of a sequence typically refers to the set of indices for which the sequence values are defined. Sequences are generally defined for all natural number indices (1, 2, 3, 4, ...). Assuming the sequence continues infinitely following the pattern, its domain is indeed all natural numbers. This statement is correct.
5. [tex]\((4, 9)\)[/tex] lies on the graph of the sequence.
In the graph of a sequence, the point [tex]\((n, f(n))\)[/tex] represents the [tex]\(n\)[/tex]-th term of the sequence. Checking this:
[tex]\[
f(4) = 9
\][/tex]
So, the point [tex]\((4, 9)\)[/tex] does lie on the graph of the sequence. This statement is correct.
To summarize the analysis:
- The sequence has 5 terms: misleading (sequence likely infinite).
- The 4th term of the sequence is 9: True.
- [tex]\( f(5) = 2 \)[/tex]: False.
- The domain of the sequence is all natural numbers: True.
- [tex]\((4,9)\)[/tex] lies on the graph of the sequence: True.
So, the correct statements among the given options are:
1. The 4th term of the sequence is 9.
4. The domain of the sequence is all natural numbers.
5. [tex]\((4, 9)\)[/tex] lies on the graph of the sequence.
