Complete the recursive formula of the geometric sequence

Question

$$0.2,-1,5,-25,\text{…}$$.
$$a(1)=$$
$$a(n)=a(n-1)\cdot$$

Answer 1

To find the recursive formula of the given geometric sequence $0.2,-1,5,-25,\dots$, let's first identify the first term and the common ratio.

1. The first term is: $$a(1)=0.2$$

2. To find the common ratio, we can divide any term by the previous term:

   • To find the ratio between the second term and the first term: $$r=\frac{-1}{0.2}=-5$$

   • To verify, we can check the ratio between the third term and the second term: $$r=\frac{5}{-1}=-5$$

   • Finally, let's check the ratio between the fourth term and the third term: $$r=\frac{-25}{5}=-5$$

Since the ratio is consistent across the terms, we confirm the common ratio $r=-5$.

Putting this information together, we can write the recursive formula for the sequence:

• The first term: $$a(1)=0.2$$

• The recursive relationship: $$a(n)=a(n-1)\cdot (-5)$$

Therefore, the completed recursive formula is:

$$\begin{array}{rlr}a(1)& =0.2,a(n)& =a(n-1)\cdot (-5)\text{ for }n>1.\end{array}$$