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