Given:
[tex]h(1)=9[/tex]
[tex]h(n)=h(n-1)\cdot (-3)[/tex]
To find:
The explicit formula for h(n).
Solution:
We have,
[tex]h(n)=h(n-1)\cdot (-3)[/tex] ...(i)
It is the recursive formula of a geometric sequence. It is of the form
[tex]a(n)=a(n-1)\cdot r[/tex] ...(ii)
where r is the common ratio.
On comparing (i) and (ii), we get
[tex]r=-3[/tex]
We have, [tex]h(1)=9[/tex] so the first term of the geometric sequence is [tex]a=9[/tex].
The explicit formula for a geometric sequence is:
[tex]h(n)=ar^{n-1}[/tex]
Substitute a=9 and r=-3 to get the explicit formula for the given sequence.
[tex]h(n)=9(-3)^{n-1}[/tex]
Therefore, the required explicit formula is [tex]h(n)=9(-3)^{n-1}[/tex].
