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

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 $f(n)=f(n-1)+2$.

Answer 2

F(n) = 2(n+2)

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