Summation in Sheets

I'm having trouble generating a mathematical summation in Google Sheets; specifically, assuming I have inputs in cells, we'll call them E2 and F2, I'm trying to perform the sum from i = E2 to i = E2+F2 of i. If LaTeX goes through, that's:

\sum_{i=E2}^{E2+F2}i

or

rendered in QuickLaTeX

Particularly I'm trying to get this into one box, because I need to do it a lot of times - that '2' is going to be most vertical values of my sheet. Unsurprisingly, this formula will go in G2.

So far I've tried a few SERIESSUM formulae, but I'm still not sure how to convert a linear series into a power series - if that's even possible.

Edit:

I originally misunderstood the question.
If I understand correctly, what you're searching for is:

=(E2+1)+(E2+2)+(E2+3)+...+(E2+F2)

(I think the original question had an E2+0 term also, that's why the formula you tried in the comments was wrong)

you can pull out E2 from each term and you'll get =F2*E2+1+2+3...+F2

So the final formula is
=E2*F2+F2*(F2+1)/2

Original answer:

If I understood your question right, you want to find the sum of numbers from the value in E2 to the value in F2.

The standard formula is =(F2-E2+1)*(E2+F2)/2

In short: multiply the number of elements between E2 and F2 by the average of the two endpoints

Try this out:


=LET(Σ,LAMBDA(first,last,expression,ARRAYFORMULA(SUM(MAP(SUBSTITUTE(expression,"n",SEQUENCE(last-first+1)),LAMBDA(_,INDEX(QUERY(,"select "&_),2)))))),Σ(E2,F2,"n"))


Although the solution is already found, I can suggest another one which is more "sheets-specific":
=SUM(SEQUENCE(F2; 1; E2 + 1))

Sometimes we don't know a ready-to-use summation formula or the required formula does not exist et all. Then we may use a more general approach, i.e. (1) generate a sequence and (2) calculate a sum of members. Google sheets have a direct way to do it and we should know about it.