Answer:
r = 9 and m = 112
Step-by-step explanation:
[tex]\sum_{k=1}^{225}W_{k}=\sum_{k=0}^{m}4r^{k}[/tex]
Write W in terms of U and V.
[tex]\sum_{k=1}^{225}(U_{k}+V_{k})=\sum_{k=0}^{m}4r^{k}\\\sum_{k=1}^{225}U_{k}+\sum_{k=1}^{225}V_{k}=\sum_{k=0}^{m}4r^{k}[/tex]
Define U and V using geometric series formula.
[tex]\sum_{k=1}^{225}2(3)^{k-1}+\sum_{k=1}^{225}2(-3)^{k-1}=\sum_{k=0}^{m}4r^{k}[/tex]
Use sum of geometric series formula.
[tex]2(\frac{1-(3)^{225}}{1-3})+2(\frac{1-(-3)^{225}}{1-(-3)})=4(\frac{1-(r)^{m+1}}{1-r})[/tex]
Simplify.
[tex]-1(1-3^{225})+\frac{1+3^{225}}{2}=4(\frac{1-(r)^{m+1}}{1-r})\\-1+3^{225}+\frac{1}{2}+\frac{3^{225}}{2}=4(\frac{1-(r)^{m+1}}{1-r})\\-\frac{1}{2}+\frac{3(3^{225})}{2}=4(\frac{1-(r)^{m+1}}{1-r})\\\frac{-1+3(3^{225})}{2}=4(\frac{1-(r)^{m+1}}{1-r})\\\frac{-1+3^{226}}{2}=4(\frac{1-(r)^{m+1}}{1-r})\\4\frac{-1+3^{226}}{8}=4(\frac{1-(r)^{m+1}}{1-r})\\4\frac{1-3^{226}}{-8}=4(\frac{1-(r)^{m+1}}{1-r})\\4\frac{1-9^{113}}{1-9}=4(\frac{1-(r)^{m+1}}{1-r})[/tex]
Therefore, r = 9 and m = 112.
