How to get Sum of array of integers?

Question

This question seems easy but I am struggling with it now, I tried a few ways and check this solution as well but couldn't help me, please check and help if you are able:
So I need to save the sum of array elements in single variable and I tried these approaches, I show you two of them here. (I must have x as variable to store result, and initialize the array as well)

Array [A, 3]
A = {1, 2, 5}
R2 = x == Sum[A[index], {index, i, 3}]  && i == 1
Simplify[Reduce[R2]]

And Also

R2 = x == Sum[A[index], {index, i, 3}] && A[1] == 1 && A[2] == 2 && A[3] == 5 && i == 1
Simplify[Reduce[R2]]

this is part of final answer I get

x == A[1] + A[2] + A[3]

It shows me a simple equation not final answer, but I expect a simple answer x== 8 or x-> 8, but it does not solve the SUM, seems I am missing something.

EDITED: based on @thorimur answer I completed my question. Actually the reason i used Reduce and Sum is: 1- I used reduce because we have more complicated expressions which Sum is part of that. 2- I used sum because not all the time we have initialized array. For some times, i need parametric array which must be shown in Sum. this is the more complete statement.

R2 = x == Sum[A[index], {index, i, 3}]  && i == 1
R3=Simplify[Reduce[
  Exists[{i,x,A},  R2 && // Array initialization // i == 1 ]]]

I know this does not work but this is my intention. we need a statement as R3

Answer 1

A couple things:

• Mathematica generally doesn't require you to initialize arrays before constructing them or anything like that; you can just construct them! Array[A, 3] does not give any definitions to the variable A; it merely outputs {A[1], A[2], A[3]} without changing any variables. Only the next line does anything to affect the symbol A, and it defines A as {1,2,5}.
• Single-brackets are for function application. So, in the above Sum, it expects A to be a function that takes in index. To access parts of a list, use double brackets. E.g. with A = {1,2,5}, A[[3]] evaluates to 5.
• Sum is typically used for mathematical sums, like $\sum_{n=1}^\infty \frac{1}{n^2}$, which would simply be written as Sum[1/n^2, {n, 1, Infinity}]. In the above, you could have simply replaced i with its value. It's not used for adding up elements of lists, generally.
• Reduce is essentially for solving equations, like Reduce[x^2 == 1, x]—you wouldn't expect to use Reduce in manipulating data structures, like you want to do here.
• Note that the solution you linked is about getting a list of successive sums, not simply adding up the elements of the list! The function Total is used for that.

So, the following is sufficient:

A = {1, 2, 5}
x = Total[A]

There are also several ways to add up a range of indices but not all of them. One of them is indeed Sum[A[[index]], {index, indexmin, indexmax}], but you could also simply do Total[A[[indexmin ;; indexmax]]]; ;; specifies a span of indices (inclusive). So, for example, if A = {1,2,5}, A[[2;;3]] evaluates to {2,5}.

Also, if you have a list of indices—say you want to add up the elements at indices indexlist = {2, 4, 8}—you can simply feed a whole list to [[ ]] and get out a list of the elements at those indices, then apply Total. E.g. indexlist = {2, 4, 8}, A[[indexlist]] gets you {A[[2]], A[[4]], A[[8]]}. Then you can Total that list.

To create a symbolic array, though, you'll want to be sure to give your array and its components different names. For example, A0 = Array[A, 3]. Note that A0[[i]] now evaluates to A[i]. I'm still not totally sure what you need, but note that you might not need to do any initialization at all. I could write Sum[A[i], {i, 1, 3}] and get A[1] + A[2] + A[3] without needing to define an array A0 of these A anywhere. Mathematica's symbolic nature is a strength here.

Hope this helps; let me know if it doesn't quite get you where you need to be, or if anything I said is unclear! :)