whats the rule for my table?

whats the rule for my table?

-4  6

 4  4

 8  3

12 2

and what is an equation for the rule?

$$\small\text{ \begin{array}{c|c|c} \hline n & a_n & \small{\text{difference (d)}} \\ \hline 1 & 16 & \\ & & -4\\ 2 & 12 \\ & & -4\\ 3 & 8 \\ & & -4\\ 4 & 4 \\ & & -4\\ 5 & 0 \\ & & -4\\ 6 & -4 \\ \hline \end{array} }}$$

$$\boxed{ a_n = a_1 + (n-1)*d } \small{\text{ arithmetic series }}$$

$$a_1 = 16 \small{\text{ and }} d = -4\qquad a_n = 16 + (n-1)*(-4) \\ a_n = 16 - 4n + 4\\ a_n = 20 - 4n \\ \small{\text{The equation for }} a_n \small{\text{ is }} \textcolor[rgb]{1,0,0}{a_n = 20 - 4n } \\ \\ n? \qquad a_n = 20 - 4n \\ 4n = 20 - a_n \quad | \quad : 4 \\ n = 5 - \frac{1}{4}a_n \\ \small{\text{The equation for }} n \small{\text{ is }}\textcolor[rgb]{1,0,0}{5 - \frac{1}{4}a_n }$$

The formula for slope is:  m  =  (y2 - y1) / (x2 - x1)

The slope from (-4, 6) to (4, 4) is:  m  =  (4 - 6) / (4 - -4)  =  -2/8  =  -1/4

The slope from (4, 4) to (8, 3) is:  m  =  (3 - 4) / (8 - 4)  =  -1/4 

The slope from (8, 3) to (12, 2) is:  m  =  (2 - 3) / (12 - 8)  =  -1/4

Since the slope is always the same, these points fall on a straight line.

The point-slope formula for a straight line is:  y - y1  =  m(x - x1)

Using (x1, y1) = (-4, 6) and m = -1/4:          y - 6  =  (-1/4)(x - -4)

                                                                       y - 6  =  (-1/4)(x + 4)

                                                                        y - 6  =  (-1/4)x - 1

                                                                        y  =  (-1/4)x + 5

Melody, you are correct, so I'll modify my answer.

I think it is just

y= -1/4x+5

If geno3141 is correct, wouldn't it simplify further to y  =  (-1/4)x + 11?

thank you guys sooo much got er' down! thanks again!

First let me test the rule:

$${\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}\left(-{\mathtt{4}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{6}}$$

$${\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}\left({\mathtt{4}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{4}}$$

$${\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}\left({\mathtt{8}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{3}}$$

$${\mathtt{\,-\,}}{\mathtt{0.25}}{\mathtt{\,\times\,}}\left(-{\mathtt{4}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{5}} = {\mathtt{6}}$$

It looks like Melody is accurate.