The Discrete Rado Number for $x_1+x_2+\cdots +x_m+c=2x_0$

For a positive integer $m$ and a real number $c$, let $R=R(m,c,2)$ denote the discrete 2-color Rado number for the equation $x_1 + x_2 + \dots + x_m + c = 2x_0$\mathrm{.}Inotherwords,$R$ is the smallest integer such that for any coloring of the integers $1,2,\dots ,R$, there exist numbers $x_1, x_2, \dots, x_m, x_0$,allwiththesamecolor,suchthat$x_1 + x_2 + \dots + x_m + c = 2x_0$. In this article we show that if $m\ge 2$ and $c>0$, then $$\begin{array}{c}R(m,c,2)=\{\begin{array}{ll}\mathrm{\infty }& \text{for }m\text{ even, }c\text{ odd}\\ \lceil \frac{m}{2}\lceil \frac{m+c}{2}\rceil +\frac{c}{2}\rceil & \text{otherwise.}\end{array}\end{array}$$ For real numbers $a$ and $c$, we look at the 2-color Rado number for the equation $x_1+c=a{x}_0$. We show that if $a>1$ and $c>0$, then the 2-color continuous Rado number is $$R_{\mathbb{R}}(1,c,a)=\{\begin{array}{ll}\mathrm{\infty }& \text{if }a=1\\ \frac{c}{a-1}& \text{otherwise.}\end{array}$$ From this, we will show that the discrete Rado number is $$R(1,c,a)=\{\begin{array}{ll}\frac{c}{a-1}& \text{if }(a-1)\mid c\\ \mathrm{\infty }& \text{otherwise.}\end{array}$$