How to sum two columns and save the values to third column using Linux shell command

I have a sum.csv file I want to sum column 1 contents and column 2 contents and save the results to column 3, let say input is

1,2
3,4
5,6

Required output is

1,2,3
3,4,7
5,6,11

I am using the command awk

awk -F "," '{$3=$1+$2}{print $3}'>>"sum.csv" sum.csv

It will create output in my sum.csv file as

1,2
3,4
5,6
3
7
11

But I want:

1,2,3
3,4,7
5,6,11

How can I get the output please guide me, also it should save on same file.

>Solution :

You can use

awk -F, '{print $0 OFS $1+$2}' OFS=, file > newfile
awk 'BEGIN{FS=OFS=","} {print $0 OFS $1+$2}' file > newfile
awk -F, '$0=$0FS$1+$2' file > newfile

See an online demo.

With -F,/OFS=, (or BEGIN{FS=OFS=","}) you set the input and output field separator to a comma, and with print $0 OFS $1+$2 you output the line plus the comma and the sum of the two filed values.