Answer
The common ratio is -$\frac{1}{2}$ so yes, this is a geometric sequence.
The explicit formula is $a_{n}$=200$\times$ $(-{\frac{1}{2})}^{n-1}$.
The recursive formula is $a_{1}$=200;$a_{n}$=$a_{n-1}$ $\times$ -$\frac{1}{2}$
Work Step by Step
You are given the sequence 200,-100,50,-25.The starting value $a_{1}$=200.Find the common ratio by using the formula: R=$\frac{a2}{a1}$,R=$\frac{a4}{a3}$.Plug in the values to get the ratio:
r=$\frac{-100}{200}$=$\frac{-1}{2}$
r=$\frac{-25}{50}$=$\frac{-1}{2}$
There is a common ratio, r=-$\frac{1}{2}$. So the sequence is geometric.
Substitute a1 and R into the explicit formula($a_{n}$=$a_{1}$ $\times$ $r^{n-1}$).The explicit formula is $a_{n}$=200$\times$ $({\frac{-1}{2}})^{n-1}$.
Substitute a1 and r into the recursive formula ($a_{1}$=A;$a_{n}$=$a_{n-1}$$\times$R). The recursive formula is $a_{1}$=200;$a_{n}$=$a_{n-1}$ $\times$ $\frac{-1}{2}$
