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#Bash + GNU utilities, 32 24

tac|nl -s*|paste -sd+|bc

Input read from STDIN.

Update: I see the input may be given as a list of digits. My input list is newline-delimited.

Try it online.

###Explanation

tac                       # reverse digit list
   |nl -s*                # prefix line numbers; separate with "*" operator
          |paste -sd+     # join lines onto one line, separated with "+" operator
                     |bc  # arithmetically evaluate