Answer:
[tex]a_{n} =\frac{-a_{n-1}}{4}[/tex].
Step-by-step explanation:
We are given a geometric sequence { -16, 4, -1, .... }
i.e. [tex]a_{1} =-16[/tex], [tex]a_{2} =4[/tex], [tex]a_{3} =-1[/tex], ...
We will first find the common ratio 'r'.
Now, [tex]r=\frac{a_{n}}{a_{n-1}}[/tex]
i.e. [tex]r=\frac{a_{2}}{a_{1}}[/tex]
i.e. [tex]r=\frac{4}{-16}[/tex]
i.e. [tex]r=\frac{1}{-4}[/tex]
Similarly, i.e. [tex]r=\frac{a_{3}}{a_{2}}[/tex]
i.e. [tex]r=\frac{-1}{4}[/tex]
So, we get that the common ratio is [tex]r=\frac{-1}{4}[/tex].
Now, the recursive formula for the geometric sequence is given by,
[tex]a_{n} =r \times a_{n-1}[/tex]
i.e. [tex]a_{n} =\frac{-1}{4} \times a_{n-1}[/tex]
i.e. [tex]a_{n} =\frac{-a_{n-1}}{4}[/tex].
Hence, the recursive formula for this sequence is [tex]a_{n} =\frac{-a_{n-1}}{4}[/tex].
