Bash script - variables in curly braces

Variables won't expand inside brace expansion. You could do:

for ((number=startNumber; number<=endNumber; number++)); do
    echo "$number"
done

Also, there is no reason to use arithmetic expansion for startNumber you should simply do:
startNumber=1.

Additionally, you don't need to use $ to expand variables inside arithmetic expansion, so endNumber could be:
endNumber=$((startNumber+3))

As explained elsewhere the expansion won't work. Alternative way to get your sequence of numbers:

for number in $(seq $startNumber $endNumber)
do
  echo $number
done

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