[tex]\underline{\ x|\ \ 0\ \ |\ \ 1\ \ |\ \ 2\ \ |\ \ 3\ \ |\ \ 4\ \ |\ \ 5\ \ |}\\U|500\ |\ 255|\ 130|\ 66\ \ |\ 34\ |\ \ 17\ |\\\\U=a(b)^x\\\\for\ x=0,\ U=500\\\\500=a(b)^0\\\\500=a(1)\to \boxed{a=500}\\\\for\ x=1,\ U=255\\\\255=500(b)^1\\\\255=500b\qquad\text{divide both sides by 500}\\\\b=\dfrac{255}{500}\\\\b=\dfrac{255:5}{500:5}\\\\b=\dfrac{51}{100}\to \boxed{b=0.51}\\\\\text{Therefore we have the equation of the function:}\\\\U=500(0.51)^x[/tex]
[tex]\text{Check for other values of x:}\\\\for\ x=2\\\\U=500(0.51)^2=130.05\approx130\qquad CORRECT\\\\for\ x=3\\\\U=500(0.51)^3=66.3255\approx66\qquad CORRECT\\\\for\ x=4\\\\U=500(0.51)^4=33.826\approx34\qquad CORRECT\\\\for\ x=5\\\\U=500(0.51)^5=17.25125\approx17\qquad CORRECT[/tex]
[tex]Answer:\ \boxed{U=500(0.51)^x}[/tex]
