how do access previous item in list using linQ?

Ok so getting the Next item in the list you can use:

A.SkipWhile(x => x != value).Skip(1).FirstOrDefault();

So to get the previous item use:

var B = A.ToList();
B.Reverse();
B.SkipWhile(x => x != value).Skip(1).FirstOrDefault();

Well, a straightforward translation would be:

var m = Enumerable.Range(1, A.Count - 1)
                  .Select(i => A[i] + A[i - 1])
                  .ToList();

But also consider:

var m = A.Skip(1)
         .Zip(A, (curr, prev) => curr + prev)
         .ToList();

Or using Jon Skeet's extension here:

var m = A.SelectWithPrevious((prev, curr) => prev + curr)
         .ToList();

But as Jason Evans points out in a comment, this doesn't help all that much with readability or brevity, considering your existing code is perfectly understandable (and short) and you want to materialize all of the results into a list anyway.

There's nothing really wrong with:

var sumsOfConsecutives = new List<int>();

for(int i = 1; i < A.Count; i++)
   sumsOfConsecutives.Add(A[i] + A[i - 1]);