$$\begin{align*} &\quad \;\;\; S = \sum_{{\color{red}{i=0}}}^{n}\sum_{{\color{blue}{j=0}}}^{i} a_{\color{red}{i}}.a_{\color{blue}{j}} \quad \dots \dots (1) \\ \end{align*}$$
From above figure,
$$\begin{align*} &\Rightarrow S = \sum_{{\color{blue}{j=0}}}^{n}\sum_{{\color{red}{i=j}}}^{n} a_{\color{red}{i}}.a_{\color{blue}{j}} \\ \end{align*}$$
Now, $a_j.a_i = a_i.a_j$
$$\begin{align*} &\Rightarrow S = \sum_{{\color{blue}{j=0}}}^{n}\sum_{{\color{red}{i=j}}}^{n} a_{\color{blue}{j}}.a_{\color{red}{i}} \\ \end{align*}$$
Now with change of varibales,
$$\begin{align*} &\Rightarrow S = \sum_{{\color{red}{i=0}}}^{n}\sum_{{\color{blue}{j=i}}}^{n} a_{\color{red}{i}}.a_{\color{blue}{i}} \quad \dots \dots (2)\\ \end{align*}$$
Adding,
$$\begin{align*} &(1)+(2) \rightarrow \\ &\Rightarrow 2S = \sum_{i=0}^{n}\left [ \sum_{j=0}^{i}a_i.a_j \;\; + \;\; \sum_{j=i}^{n}a_i.a_j \right ] \\ &\Rightarrow 2S = \sum_{i=0}^{n}\left [ \sum_{j=0}^{n}a_i.a_j \;\; + \;\; a_i^2 \right ] \\ &\Rightarrow 2S = \sum_{i=0}^{n}\sum_{j=0}^{n}a_i.a_j \;\; + \;\; \sum_{i=0}^{n}a_i^2 \\ &\Rightarrow 2S = \left ( \sum_{i=0}^{n}a_i \right )\left ( \sum_{j=0}^{n}a_j \right ) \;\; + \;\; \sum_{i=0}^{n}a_i^2 \\ &\Rightarrow S = \frac{1}{2}.\left [ \left ( \sum_{i=0}^{n}a_i \right )^2 \;\; + \;\; \left ( \sum_{i=0}^{n}a_i^2 \right ) \right ] \\ \end{align*}$$