var=$((var + 1))

Arithmetic in bash uses $((...)) syntax.

Various options to increment by 1, and performance analysis

Thanks to Radu Rădeanu's answer that provides the following ways to increment a variable in bash:

var=$((var+1))
((var=var+1))
((var+=1))
((var++))
let "var=var+1"
let "var+=1" 
let "var++"

There are other ways too. For example, look in the other answers on this question.

let var++
var=$((var++))
((++var))
{
    declare -i var
    var=var+1
    var+=1
}
{
    i=0
    i=$(expr $i + 1)
}

Having so many options leads to these two questions:

1. Is there a performance difference between them?
2. If so which, which performs best?

Incremental performance test code:

#!/bin/bash

# To focus exclusively on the performance of each type of increment
# statement, we should exclude bash performing while loops from the
# performance measure. So, let's time individual scripts that
# increment $i in their own unique way.

# Declare i as an integer for tests 12 and 13.
echo > t12 'declare -i i; i=i+1'
echo > t13 'declare -i i; i+=1'
# Set i for test 14.
echo > t14 'i=0; i=$(expr $i + 1)'

x=100000
while ((x--)); do
    echo >> t0 'i=$((i+1))'
    echo >> t1 'i=$((i++))'
    echo >> t2 '((i=i+1))'
    echo >> t3 '((i+=1))'
    echo >> t4 '((i++))'
    echo >> t5 '((++i))'
    echo >> t6 'let "i=i+1"'
    echo >> t7 'let "i+=1"'
    echo >> t8 'let "i++"'
    echo >> t9 'let i=i+1'
    echo >> t10 'let i+=1'
    echo >> t11 'let i++'
    echo >> t12 'i=i+1'
    echo >> t13 'i+=1'
    echo >> t14 'i=$(expr $i + 1)'
done

for script in t0 t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14; do
    line1="$(head -1 "$script")"
    printf "%-24s" "$line1"
    { time bash "$script"; } |& grep user
    # Since stderr is being piped to grep above, this will confirm
    # there are no errors from running the command:
    eval "$line1"
    rm "$script"
done