Expanding n-th power of a sum of N numbers

Question

Consider a sum (finite) of $N-1$ terms: $$\sum_{i=1}^{N-1}a_i$$

I can show, $$\left(\sum_{i=1}^{N-1}a_i\right)^2 = \sum_{i=1}^{N-1}a^2_i + 2\sum_{i'<i}^{N-1}a_ia_{i'}$$

for a problem I am working on, I need the result $$\left(\sum_{i=1}^{N-1}a_i\right)^4$$

Is there a general theorem/formula that can give us. Could someone point me to some references $$\left(\sum_{i=1}^{N-1}a_i\right)^n$$

Answer 1

$$\left(\sum_{i=1}^{N-1}a_i\right)^4=\bigg(\left(\sum_{i=1}^{N-1}a_i\right)^2 \bigg)^2$$

$$=\bigg (\sum_{i=1}^{N-1}a^2_i + 2\sum_{i'<i}^{N-1}a_ia_{i'}\bigg )^2$$

$$=\bigg (\sum_{i=1}^{N-1}a^2_i\bigg)^2 + 4 \bigg (\sum_{i'<i}^{N-1}a_ia_{i'}\bigg )^2$$

$$+4\sum_{i=1}^{N-1}a^2_i\sum_{i'<i}^{N-1}a_ia_{i'}$$