Answer:
x₃ = 62
Step-by-step explanation:
A recursive rule for a sequence is a formula that generates each term based on the previous term. In a recursive definition, we need to also define the first term (x₁) because it serves as the starting point for the sequence, and the recursive formula relies on it to generate subsequent terms.
Given sequence:
[tex]\begin{cases}x_{n+1} = 3x_n + 2\\x_1=6\end{cases}[/tex]
To find the value of x₃ in the given recursive sequence, we first need to find the value of x₂ by substituting x₁ = 6 into the given rule:
[tex]\begin{aligned}x_2 &= 3x_1 + 2\\x_2 &= 3(6) + 2\\x_2 &= 18 + 2\\x_2 &= 20\end{aligned}[/tex]
Now, we can find the value of x₃ by substituting x₂ = 20 into the same rule:
[tex]\begin{aligned}x_3 &= 3x_2 + 2\\x_3 &= 3(20) + 2\\x_3 &= 60 + 2\\x_3 &= 62\end{aligned}[/tex]
Therefore, the value of x₃ in the given sequence is 62.
