The Modern Way
I personally recommend you use unicode-math
on LuaLaTeX or XeLaTeX when you can, and the legacy toolchain when you have to. You’ll get more symbols with more consistency than any combination of legacy packages could give you, it’ll be simpler, and you’ll even be able to copy and paste from the PDF.
It also allows you to use any system font in LaTeX, including Arial from the MS core fonts.
\documentclass[ngerman]{scrartcl}
\usepackage{amsmath}
\usepackage{unicode-math}
\usepackage{babel}
\defaultfontfeatures{ Scale=MatchLowercase, Ligatures = TeX }
\setmainfont{Arial}
\setsansfont{Arial}
\setmonofont{Andale Mono}
\setmathfont{GFSNeohellenicMath.otf}
\setmathfont[range=up]{Arial}
\setmathfont[range=it]{Arial Italic}
\setmathfont[range=bfup]{Arial Bold}
\setmathfont[range=bfit]{Arial Bold Italic}
\setmathfont[range=tt]{Andale Mono}
\newcommand\EUR[1]{\mbox{#1\,\texteuro}}
\usepackage[locale=DE]{siunitx}
\begin{document}
\begin{minipage}{10cm}
Text.
\begin{align}
K & = A_0 + \sum\limits_{t = 1}^{T} A_t ( 1 + i )^{-t} \text{, mit $T = 3$ und } i = \SI{4}{\%} = \frac{4}{100} \Rightarrow\\
& = \left( - 500 + \ldots + \frac{200}{1{,}04^3}\right) \EUR{1000}
\end{align}
Text.
\end{minipage}
\end{document}

With Modest Changes
This version adapts your MCVE with the following adjustments: It explicitly sets up textcomp
to use the Arial version of the Euro symbol (and declares \EUR
for compatibility), sets the math symbols to the sans-serif version of newtx, and sets the math letters to the text font with mathastext
.
\documentclass[ngerman]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % The default since 2018
\usepackage{
lmodern,
babel,
newtxsf,
uarial,%from https://tex.stackexchange.com/questions/23957/how-to-set-font-to-arial-throughout-the-entire-document/23961#23961
textcomp,
amsmath
}
%from https://tex.stackexchange.com/questions/23957/how-to-set-font-to-arial-throughout-the-entire-document/23961#23961
\renewcommand{\familydefault}{\sfdefault}
\usepackage[italic]{mathastext}
\newcommand\EUR[1]{\mbox{#1\,\texteuro}}
\usepackage[locale=DE]{siunitx}
\begin{document}
\begin{minipage}{10cm}
Text.
\begin{align}
K & = A_0 + \sum\limits_{t = 1}^{T} A_t ( 1 + i )^{-t} \text{, mit $T = 3$ und } i = \SI{4}{\%} = \frac{4}{100} \Rightarrow\\
& = \left( - 500 + \ldots + \frac{200}{1{,}04^3}\right) \EUR{1000}
\end{align}
Text.
\end{minipage}
\end{document}

It is also possible to use Microsoft’s version of Arial in PDFTeX through the winfonts
package. Use the command \fontfamily{arial}\selectfont
, or redefine \familydefault
and perhaps \sfdefault
.
\documentclass[ngerman]{scrartcl}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc} % The default since 2018
\usepackage{
lmodern,
babel,
newtxsf,
winfonts,
textcomp,
amsmath
}
\renewcommand{\sfdefault}{arial}
\renewcommand{\familydefault}{\sfdefault}
\usepackage[italic]{mathastext}
\newcommand\EUR[1]{\mbox{#1\,\texteuro}}
\usepackage[locale=DE]{siunitx}
\begin{document}
\begin{minipage}{10cm}
Text.
\begin{align}
K & = A_0 + \sum\limits_{t = 1}^{T} A_t ( 1 + i )^{-t} \text{, mit $T = 3$ und } i = \SI{4}{\%} = \frac{4}{100} \Rightarrow\\
& = \left( - 500 + \ldots + \frac{200}{1{,}04^3}\right) \EUR{1000}
\end{align}
Text.
\end{minipage}
\end{document}

Including the winfonts
package isn’t strictly necessary here, but at least requiring it tells the user what package needs to be installed.
\cdots
instead of\ldots
,;-)
) – Ruixi Zhang Jan 13 '19 at 3:26