Sigma/Sum transformation If you have
$$\sum_{j=2}^{n} c_{j} \times a_{j} + \left(1-\sum_{j=2}^{n}c_j\right)a_{1} $$
How can you transform it into
$$ a_{1}+ \sum_{j=2}^{n} \left(c_j \times (a_{j}-a_{1})\right)$$
I know the basics of sum calculations, but I don't understand, how to transform it. Especially with the 1-Sigma term. Can someone explane it step by step ?

\begin{align}
\sum_{j=2}^{n} c_{j} \times a_{j} + \left(1-\sum_{j=2}^{n}c_j\right)a_{1} &= \left(\sum_{j=2}^{n} c_{j} \times a_{j} \right) + a_{1} - \sum_{j=2}^{n}c_{j}a_{1}\\
&=a_{1} + \sum_{j=2}^{n}c_{j}a_{j} - \sum_{j=2}^{n}c_{j}a_{1}\\
&=a_{1} + \sum_{j=2}^{n}c_{j}(a_{j} - a_{1})
\end{align}
