[tex]a_{n+1}[/tex] = - 3 [tex]a_{n}[/tex] with [tex]a_{1}[/tex] = 4
To find the next term in the sequence multiply the previous term by - 3
the common ratio r = [tex]\frac{-108}{36}[/tex] = [tex]\frac{36}{-12}[/tex] = - 3
[tex]a_{n+1}[/tex] = [tex]a_{n}[/tex] × r ← recursive formula
Answer: [tex]a_{n+1}=-3a_{n}[/tex]
Step-by-step explanation:
The recursive formula for geometric sequence is given by :-
[tex]a_{n+1}=a_{n}r[/tex] -----(1) , where r = common ratio and n=natural number .
nth term of geometric sequence = [tex]ar^{n-1}[/tex]
The given geometric sequence : 4,-12,36,-108...
First term = [tex]a_1=a=4[/tex]
Second term = [tex]a_2=ar=-12[/tex]
Also, [tex]r=\dfrac{ar}{a}=\dfrac{-12}{4}=-3[/tex]
∴ r = -3
Put the value of r in (1) , we get the recursive formula for given geometric sequence as
[tex]a_{n+1}=a_{n}(-3)[/tex]
i.e. [tex]a_{n+1}=-3a_{n}[/tex]
