Question

find the common ratio and zeroth term, then write the equation. geometric equations (t(n)=t(0)cdot r^{n}) sequence: 200, 100, 50, 25, ...; 4, -12, 36, -108, ...; 2, 4, 8, 16, ... common ratio zeroth term equation

Explanation:

Step1: Calculate common ratio for 200, 100, 50, 25, ...

Divide second - term by first - term: $r=\frac{100}{200}=\frac{1}{2}$

Step2: Find zeroth term for 200, 100, 50, 25, ...

$t(0)=\frac{t(1)}{r}=\frac{200}{\frac{1}{2}} = 400$

Step3: Write geometric equation for 200, 100, 50, 25, ...

$t(n)=400\cdot(\frac{1}{2})^n$

Step4: Calculate common ratio for 4, - 12, 36, - 108, ...

Divide second - term by first - term: $r=\frac{-12}{4}=-3$

Step5: Find zeroth term for 4, - 12, 36, - 108, ...

$t(0)=\frac{t(1)}{r}=\frac{4}{-3}=-\frac{4}{3}$

Step6: Write geometric equation for 4, - 12, 36, - 108, ...

$t(n)=-\frac{4}{3}\cdot(-3)^n$

Step7: Calculate common ratio for 2, 4, 8, 16, ...

Divide second - term by first - term: $r = \frac{4}{2}=2$

Step8: Find zeroth term for 2, 4, 8, 16, ...

$t(0)=\frac{t(1)}{r}=\frac{2}{2}=1$

Step9: Write geometric equation for 2, 4, 8, 16, ...

$t(n)=1\cdot(2)^n=2^n$

Answer:

Sequence	Common Ratio	Zeroth Term	Equation
4, - 12, 36, - 108, ...	- 3	$-\frac{4}{3}$	$t(n)=-\frac{4}{3}\cdot(-3)^n$
2, 4, 8, 16, ...	2	1	$t(n)=2^n$