How to write a closed form expression for $\sum \limits_{i=0}^{n-1}a_i$ in terms of $a_0$ and $\sum \limits_{i=1}^{n}a_i$?

Question

I am given this:$$\sum _{i=1}^n a_i = n^2-n,a_0=4$$ How do I write a closed form expression for $$\sum _{i=0}^{n-1}$$in terms of n?

I know that for $$\sum _{i=1}^{n-1}$$ the expression would be $a_i$=n(n-3)-2.

But for $i$=0, I am lost.

Answer 1

Without knowing what $a_0$ is, you can only say that $$\sum_{i=0}^{n-1} a_i = a_0 + \sum_{i=1}^{n-1} a_i$$

If you know that $$\sum_{i=1}^n a_i = n^2 - n$$

is true for every value of $n$, then you can continue to write $$a_0 + \sum_{i=1}^{n-1} a_i = a_0 + (n-1)^2 - (n-1)$$