adding numbers present in array without using sum function

Agree, sum() function should be your go-to for "summing" an array.  

But if you are looking to unnecessarily complicate your code, how about a recursive function to add an array of numbers.

Note with recursive rules, you must save them prior to testing.

/* 
rule!chris_add_recur(
  array: List of Integer
  index: Integer
  total: Integer
)
*/

a!localVariables(
  local!index: if(rule!APN_isEmpty(ri!index),1,ri!index),
  local!total: if(rule!APN_isEmpty(ri!total),0,ri!total),
  
  if(
    local!index>count(ri!array),
    ri!total,
    rule!chris_add_recur(
      array: ri!array,
      index: local!index+1,
      total: index(ri!array,local!index)+local!total
    )
  )
)

Call this rule with only the array input as:

rule!chris_add_recur(
  array: {1,2,3,4}
)