There is more than one way to increment a variable in bash, but what you tried will not work.

You can use, for example, arithmetic expansion:

var=$((var+1))
((var=var+1))
((var+=1))
((var++))

Or you can use let:

let "var=var+1"
let "var+=1"
let "var++"

See also: https://tldp.org/LDP/abs/html/dblparens.html.

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

Results:

i=$((i+1))              user    0m0.992s
i=$((i++))              user    0m0.964s
((i=i+1))               user    0m0.760s
((i+=1))                user    0m0.700s
((i++))                 user    0m0.644s
((++i))                 user    0m0.556s
let "i=i+1"             user    0m1.116s
let "i+=1"              user    0m1.100s
let "i++"               user    0m1.008s
let i=i+1               user    0m0.952s
let i+=1                user    0m1.040s
let i++                 user    0m0.820s
declare -i i; i=i+1     user    0m0.528s
declare -i i; i+=1      user    0m0.492s
i=0; i=$(expr $i + 1)   user    0m5.464s

Conclusion:

It seems bash is fastest at performing i+=1 when $i is declared as an integer. let statements seem particularly slow, and expr is by far the slowest because it is not a built into bash.

There's also this:

var=`expr $var + 1`

Take careful note of the spaces and also ` is not '

While Radu's answers, and the comments, are exhaustive and very helpful, they are bash-specific. I know you did specifically ask about bash, but I thought I'd pipe in since I found this question when I was looking to do the same thing using sh in busybox under uCLinux. This portable beyond bash.

If you declare $var as an integer, then what you tried the first time will actually work:

$ declare -i var=5
$ echo $var
5
$ var=$var+1
$ echo $var
6

Reference: Types of variables, Bash Guide for Beginners

There's one method missing in all the answers - bc

$ VAR=7    
$ bc <<< "$VAR+2"
9
$ echo $VAR
7
$ VAR=$( bc <<< "$VAR+1" )
$ echo $VAR
8

bc is specified by POSIX standard, so should be present on all versions of Ubuntu and POSIX-compliant systems. The <<< redirection could be altered to echo "$VAR" | bc for portability, but since the question asks about bash - it's OK to just use <<<.

The return code 1 issue is present for all default variants (let, (()), etc.). This often causes trouble, e.g., in scripts that use set -o errexit. Here is what I am using to prevent error code 1 from math expressions that evaluate to 0;

math() { (( "$@" )) || true; }

math a = 10, b = 10
math a++, b+=2
math c = a + b
math mod = c % 20
echo $a $b $c $mod
#11 12 23 3

This has to be the worst way to accomplish such a simple task but I just wanted to document it for fun I guess (complete opposite of code golf).

$ var=0
$ echo $var
0
$ var="$(python -c 'print('$var'+1)')"
$ echo $var
1

or

$ var="$(printf '%s\n' $var'+1' | bc)"
$ echo $var
1

Seriously use one of the other much better choices here.

This is the safe bet

(( var = var + 1 ))

If the resulting value is non zero, then setting exit on error will stop Your script

set -e
var=0
(( var++ ))
echo You will never get here