We had our quiz in Pre-calculus earlier and we argued about the index in writing sigma notation from a given expression.
$$1-2+3-4+5-6+\cdots -10$$
We were instructed to write the expression in sigma notation. I had my answer with $$ \sum_{n = 2}^{11}(-1)^n(n-1) $$ I was told that my answer was incorrect. So I want to ask if is it necessary for the index to be equal to 1?
On
I don't see anything wrong with your sum. It's a bit quirky and unusual to start at $n = 2$ in this case, but it's not wrong, and it's not necessary to start at $1$. (As a reality check: why would we specify that we start at $1$ the times that we do, if it's a necessity? This isn't an airtight argument, mind you. Notation can be superfluous at times.)
$$\cdots\\ \sum_{n = 0}^{9}(-1)^n(n+1)\\ \sum_{n = 1}^{10}(-1)^{n-1}n\\ -\sum_{n = 1}^{10}(-1)^nn\\ \sum_{n = 1}^{10}(-1)^{n+1}n\\ \sum_{n = 2}^{11}(-1)^n(n-1)\\ \cdots $$
are all equivalently valid notations.