How do I perform this calculation and output it to standard out?

Although you ask to do this in Bash, there's no native support for functions like square root or ceiling. It would be simpler to delegate to Perl:

perl -wmPOSIX -e "print POSIX::ceil((sqrt(1 + 8 * $n) - 1) / 2)"

Alternatively, you could use bc to calculate the square root, and some Bash to calculate the ceiling.

Let's define a function that prints the result of the formula with sqrt of bc:

formula() {
    local n=$1
    bc -l <<< "(sqrt(1 + 8 * $n) - 1) / 2"
}

The -l flag changes the scale from the default 0 to 20. This affects the scale in the display of floating point results. For example, with the default zero, 10 / 3 would print just 3. We need the floating point details in the next step to compute the ceiling.

ceil() {
    local n=$1
    local intpart=${n%%.*}
    if [[ $n =~ \.00*$ ]]; then
        echo $intpart
    else
        echo $((intpart + 1))
    fi
}

The idea here is to extract the integer part, and if the decimal part is all zeros, then we print simply the integer part, otherwise the integer part + 1, as that is the ceiling.

And a final simple function that combines the above functions to get the result that you want:

compute() {
    local n=$1
    ceil $(formula $n)
}

And a checker function to test it:

check() {
    local n num
    for n; do
        num=$(formula $n)
        echo $n $num $(compute $n)
    done
}

Let's try it:

check 1 2 3 4 7 11 12 16 17

It produces:

1 1.00000000000000000000 1
2 1.56155281280883027491 2
3 2.00000000000000000000 2
4 2.37228132326901432992 3
7 3.27491721763537484861 4
11 4.21699056602830190566 5
12 4.42442890089805236087 5
16 5.17890834580027361089 6
17 5.35234995535981255455 6

You can use bc's sqrt function.

echo "(sqrt(1 + 8 * 3) - 1) / 2" | bc

Ceil function can be implemented using any of the methods described in this answer.

Getting Ceil integer

For eg:

  ceiling_divide() {
      ceiling_result=`echo "($1 + $2 - 1)/$2" | bc`
    }

You can use bc for all the job

$>cat filebc

print "Enter a number\n";
scale=20
a=read()
b=((sqrt(1 + 8 * a) - 1) / 2)
scale=0
print "ceil = ";
((b/1)+((b%1)>0))
quit

Call it like that

bc -q filebc