We have to write a recursive formula for the sequence 6, 9, 13.5, 20.25...
A recursive formula is a formula where the term value depends on the previous term value:
[tex]a_n=f(a_{n-1})[/tex]
In this case, as there is no common difference, we can conclude that this is not a arithmetic sequence.
We will test if we have a common ratio between the terms:
[tex]\begin{gathered} \frac{a_2}{a_1}=\frac{9}{6}=1.5 \\ \frac{a_3}{a_2}=\frac{13.5}{9}=1.5 \\ \frac{a_4}{a_3}=\frac{20.25}{13.5}=1.5 \end{gathered}[/tex]
We have a common ratio between consecutive terms and its value is r=1.5.
Then we have a geometric sequence and we can write the recursive formula as:
[tex]\begin{gathered} a_n=r\cdot a_{n-1} \\ a_n=1.5\cdot a_{n-1} \end{gathered}[/tex]
You can find new terms multiplying the previous term value by 1.5.
Answer: the recursive formula is a(n) = 1.5 * a(n-1)
