Question

Formulate the recursive formula for the following geometric sequence. {-16, 4, -1, ...}

Answer 1

Answer:

$a_{n} =\frac{-a_{n-1}}{4}$.

Step-by-step explanation:

We are given a geometric sequence { -16, 4, -1, .... }

i.e. $a_{1} =-16$, $a_{2} =4$, $a_{3} =-1$, ...

We will first find the common ratio 'r'.

Now, $r=\frac{a_{n}}{a_{n-1}}$

i.e. $r=\frac{a_{2}}{a_{1}}$

i.e. $r=\frac{4}{-16}$

i.e. $r=\frac{1}{-4}$

Similarly, i.e. $r=\frac{a_{3}}{a_{2}}$

i.e. $r=\frac{-1}{4}$

So, we get that the common ratio is $r=\frac{-1}{4}$.

Now, the recursive formula for the geometric sequence is given by,

$a_{n} =r \times a_{n-1}$

i.e. $a_{n} =\frac{-1}{4} \times a_{n-1}$

i.e. $a_{n} =\frac{-a_{n-1}}{4}$.

Hence, the recursive formula for this sequence is $a_{n} =\frac{-a_{n-1}}{4}$.