Sure! Let's solve this problem step-by-step.
### Problem:
We are given the initial terms of a sequence and the rule that each term is the previous term plus 3. The given sequence is:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{term \#} & 1 & 2 & 3 & 4 \\
\hline
\text{value} & 10 & 13 & 16 & \text{?} \\
\hline
\end{array}
\][/tex]
We need to find the value of the 4th term in the sequence.
### Recursive Relation:
The relationship between consecutive terms in this sequence is given as:
[tex]\[ \text{current term} = \text{previous term} + 3 \][/tex]
### Explicit Equation:
We can also use an explicit equation to find the value of any term in the sequence directly. The explicit formula for this sequence can be written as:
[tex]\[ \text{term}_n = \text{term}_1 + (n - 1) \times 3 \][/tex]
where [tex]\( \text{term}_1 \)[/tex] is the first term of the sequence, and [tex]\( n \)[/tex] is the term number.
### Finding the 4th Term:
Given:
- First term ([tex]\(\text{term}_1\)[/tex]) = 10
- Difference between consecutive terms = 3
- We need to find the 4th term ([tex]\(n = 4\)[/tex])
Substitute these values into the explicit equation:
[tex]\[ \text{term}_4 = 10 + (4 - 1) \times 3 \][/tex]
[tex]\[ \text{term}_4 = 10 + 3 \times 3 \][/tex]
[tex]\[ \text{term}_4 = 10 + 9 \][/tex]
[tex]\[ \text{term}_4 = 19 \][/tex]
### Conclusion:
The value of the 4th term in the sequence is:
[tex]\[ \boxed{19} \][/tex]
So the value of the 4th term is 19.
