Answer:
The recursive formula defines the sequence is [tex]f(n)=f(n-1)+2[/tex].
Step-by-step explanation:
Three terms of an arithmetic sequence are
f(1) = 6
f(4) = 12
f(7) = 18
It means first term is 6, fourth term is 12 and seventh term is 18.
The formula for nth term of an arithmetic sequence is
[tex]a_n=a+(n-1)d[/tex]
where n is number of term, a is first term and d is common difference.
4th term of the AP is
[tex]a_n=6+(4-1)d[/tex]
[tex]12=6+3d[/tex] [tex][\because a_4=f(4)=12][/tex]
Subtract 6 from both sides.
[tex]12-6=3d[/tex]
[tex]6=3d[/tex]
Divide both sides by 3.
[tex]2=d[/tex]
The common difference is 2.
The recursive formula for an AP is
[tex]a_n=a_{n-1}+d[/tex]
Substitute d=2 in the above equation.
[tex]a_n=a_{n-1}+2[/tex]
It can be written as
[tex]f(n)=f(n-1)+2[/tex]
Therefore the recursive formula defines the sequence is [tex]f(n)=f(n-1)+2[/tex].
