Sigma/Sum transformation

Question

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 ?

Answer 1

\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}