Answer:
[tex]\left \{ {{a_{1} =8} \atop {a_{n} =a_{n-1} +3}} \right.[/tex]
Step-by-step explanation:
The given rule is
[tex]a_{n}=3n+5[/tex]
If we substitute [tex]n=1, n=2, n=3, n=4,..[/tex]
The sequence would be
[tex]a_{1}=3(1)+5=3+5=8\\a_{2}=3(2)+5=6+5=11\\a_{3}=3(3)+5=9+5=14\\a_{4}=3(4)+5=12+5=17[/tex]
As you can observe, the arithmetic sequence has a difference of 3, that is, adding 3 units to one term, we obtain the next one. Now, we can use this rule to right a recursive rule.
First, we have to write the first term
[tex]a_{1}= 8[/tex]
Then, we write the patter to obtain the next terms, which is adding 3 units, so
[tex]a_{n}=a_{n-1}+3[/tex]
Therefore, the recursive rule for [tex]a_{n}=3n+5[/tex] is
[tex]\left \{ {{a_{1} =8} \atop {a_{n} =a_{n-1} +3}} \right.[/tex]
