How can I sum numbers on lines in a file

bc with a little help from paste to get the lines in a single one with + as the separator:

paste -sd+ file.txt | bc

To use the output of grep (or any other command) instead a static file, pass the grep's STDOUT to the STDIN of paste:

grep .... | paste -sd+ | bc

Example:

% cat file.txt            
1
3
4
1
4
3
1
2

% paste -sd+ file.txt | bc
19

% grep . file.txt | paste -sd+ | bc
19

If you must use bash, then you can use an array to save the file contents and then iterate over the elements or you can read the file line by line and do the sum for each line, the second approach would be more efficient:

$ time { nums=$(<file.txt); for i in ${nums[@]}; do (( sum+=i )); done; echo $sum ;}
19

real    0m0.002s
user    0m0.000s
sys 0m0.000s

$ time { while read i; do (( sum+=i )); done <file.txt; echo $sum ;}
19

real    0m0.000s
user    0m0.000s
sys 0m0.000s

You could use awk, too. To count the total number of lines in *.txt files that contain the word "hello":

grep -ch 'hello' *.txt | awk '{n += $1}; END{print n}'

To simply sum the numbers in a file:

awk '{n += $1}; END{print n}' file.txt

Use numsum from the package num-utils!

(You may need to sudo apt-get install num-utils)

The command numsum does just what you need by default;

$ numsum file.txt 
19

Reading the test numbers line by line from stdin:

$ printf '
1 
3
4
1
4
3
1
2' | numsum
19

Or reading from one line:

$ printf '1 3 4 1 4 3 1 2' | numsum -r
19

More utilities

The package contains some other utilities for number processing that deserve to be more well known:

numaverage   - find the average of the numbers, or the mode or median
numbound     - find minimum of maximum of all lines
numgrep      - to find numbers matching ranges or sets
numinterval  - roughly like the first derivative
numnormalize - normalize numbers to an interval, like 0-1
numrandom    - random numbers from ranges or sets, eg odd.  
numrange     - similar to seq
numround     - round numbers up, down or to nearest

and a more general calculator command numprocess,
that applies an expression from the command line to numbers on input lines.

You can use awk, a native linux application usefull to scanning and processing files with a pattern per line. For your question, this will produce what you want:

awk 'BEGIN { sum=0 } { sum+=$1 } END {print sum }' file.txt

Pipes are also accept:

cat file.txt | awk 'BEGIN { sum=0 } { sum+=$1 } END {print sum }'

Perl solution:

$ perl -lnae '$c+=$_;END{print $c}' input.txt                                                                            
19

The above can sum all numbers across multiple files:

$ perl -lnae '$c+=$_;END{print $c}' input.txt input2.txt                                                                 
34

For multiple files given on command-line where we want to see sum of numbers in individual file we can do this:

$ perl -lnae '$c+=$_;if(eof){printf("%d %s\n",$c,$ARGV);$c=0}' input.txt input2.txt                                      
19 input.txt
15 input2.txt

A simple approach is to use a built-in feature of your shell:

SUM=0; while read N; do SUM=$((SUM+N)); done </path/to/file
echo $SUM

This reads your file linewise, sums up and prints the result.

If you want to use a pipe and only use the 1st row, it works like this:

SUM=0
your_command | while read -r LINE; do for N in $LINE; do break; done; SUM=$((SUM+N)); done
echo $SUM

Getting the first element is done like this:

LIST="foo bar baz"
for OBJ in $LIST; do break; done
echo $OBJ

foo

This is a fairly simple use of bash scripting.

SUM=0; for line in `cat file.txt`; do SUM=$((SUM + line)); done