Shell Script (Bash) to sum multiples of different arguments below a number

I'm interested in learning how to create shell scripts with bash so here's one of my first exercises, taken from here. It is an extension of the problems Elementary - 4 and 5.

The goal of the program is to sum all multiples or ARG1 or ARG2 below the user's input number.

#!/bin/bash
echo "This program will find all the multiples of ARG1 or ARG2 below or equal to your number and sum them up. Please input your limit:"
read limit

commonMultiple=`expr $1 \* $2`

sumArg1=0
sumArg2=0
sumCommonMultiple=0

for (( i = 0; i <= $limit; i+=$1 )); do
    sumArg1=`expr $sumArg1 + $i`
done
for (( i = 0; i <= $limit; i+=$2 )); do
    sumArg2=`expr $sumArg2 + $i`
done
for (( i = 0; i <= $limit; i+=$commonMultiple )); do
    sumCommonMultiple=`expr $sumCommonMultiple + $i`
done

echo "The result is `expr $sumArg1 + $sumArg2 - $sumCommonMultiple`."

This works perfectly but I'd like the input of more experienced programmers. I don't think my logic is efficient. There must be something obvious I'm missing that could have made my code a lot simpler.

Disclaimer: apparently some of the stuff I used is outdated. I'll take it into account for my future code.*

Arithmetic operations

If you look closely, something is a bit odd here:

for (( i = 0; i <= $limit; i+=$1 )); do

Ideas? How about: i is written without a dollar, but limit is written with a dollar... In fact you can drop the dollar from limit too, and the expression will be slightly simpler. From $1 you cannot drop it, as the meaning of 1 would be taken literally, because it's numeric, unlike the others.

The same is true for commonMultiple here, you can drop the dollar:

for (( i = 0; i <= $limit; i+=$commonMultiple )); do

The general rule is, in arithmetic operations within ((...)) you don't need to use the dollars in variable names.

Don't use backticks

Instead of this:

commonMultiple=`expr $1 \* $2`

This is the recommended writing style for sub shells:

commonMultiple=$(expr $1 \* $2)

Don't use expr

It's better to avoid expr, and you can do the same thing easier using Bash arithmetics, for example:

((commonMultiple = $1 * $2))

The for loops can be simplified similarly, instead of:

for (( i = 0; i <= $limit; i+=$1 )); do
    sumArg1=`expr $sumArg1 + $i`
done

You can write:

for (( i = 0; i <= limit; i+=$1 )); do
    ((sumArg1 += i))
done

Apply the same thing everywhere.

StackExchange Code Review Q#112806, answer score: 10