Answer:
[tex]\begin{cases} \sf a_1=-6 \\ \sf a_n=a_{n-1}+4 \end{cases}[/tex]
Step-by-step explanation:
Given:
- a₄ = 6
- a₅ = 10
- a₆ = 14
- a₇ = 18
- a₈ = 22
From inspection of the given terms, it is evident that there is a common difference between consecutive terms. Therefore, this is an arithmetic sequence.
A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
From inspection of the sequence, we can see that each term is found by adding 4 to the previous term. Therefore, the recursive formula is:
[tex]\sf a_n=a_{n-1}+4[/tex]
A recursive rule comprises the recursive formula together with the 1st term a₁. To find the 1st term, keep subtracting 4 from each term:
[tex]\begin{aligned} \sf a_3 & = \sf a_4 - 4\\ & = \sf 6 - 4\\ & = \sf 2\end{aligned}[/tex]
[tex]\begin{aligned} \sf a_2 & = \sf a_3 - 4\\ & = \sf 2-4 \\ & = \sf -2\end{aligned}[/tex]
[tex]\begin{aligned} \sf a_1 & = \sf a_2 - 4\\ & = \sf -2-4 \\ & = \sf -6\end{aligned}[/tex]
Therefore, the recursive rule that can be used to find the nth term of the sequence is:
[tex]\begin{cases} \sf a_1=-6 \\ \sf a_n=a_{n-1}+4 \end{cases}[/tex]
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