Question

Three terms of an arithmetic sequence are shown below. Which recursive formula defines the sequence? f(1) = 6, f(4) = 12, f(7) = 18

Answer 1

f(n) = 2(n+2)

can also be written as
f(n) = 2n + 4

Answer 2

The recursive formula defines the sequence is
`f(n)=f(n-1)+2`.

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`.