What will be the expansion of $(a_1+a_2+...a_n)^2$

Question

I have learnt binomial expansion in case of two terms, but I wondered how will one expand such whole squares, will we express it in terms of $(1+x)^2$, where $x$ can be $(a_1+a_2+...a_n) -1$

Answer 1

The formula you're thinking of is the multinomial theorem. The Wikipedia page goes pretty in depth about it and how it works.

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In addition to that, we can derive another expansion similar to that.$$\left(\sum\limits_{k=0}^{n}a_k\right)^2=\sum\limits_{k=0}^{n}a_k^2+2\sum\limits_{0\leq i < j\leq n}a_ia_j$$

I'll explain a bit what this means. The left-hand side is obviously the sum of the terms you want to expand, such as $a_0,a_1,a_2,$ etc. The right-hand side is the expansion.

The first sum is simple all the terms from the left-hand side squared. The second sum is interpreted as the sum of all of the products of the terms taken two at a time.

Some examples are$$\begin{align*}(a_0+a_1+a_2)^2 & =a_0^2+a_1^2+a_2^2+2a_0\left(a_1+a_2\right)+2a_1a_2\\ & =a_0^2+a_1^2+a_2^2+2a_0a_1+2a_0a_2+2a_1a_2\end{align*}$$$$\begin{align*}(a_0+a_1+a_2+a_3)^2 & =\sum\limits_{i=0}^{3}a_i^2+2a_0(a_1+a_2+a_3)+2a_1(a_2+a_3)+2a_2a_3\\ & =\sum\limits_{i=0}^3a_i^2+2a_0a_1+2a_0a_2+2a_0a_3+2a_1a_2+2a_1a_3+2a_2a_3\end{align*}$$

And so on...