I'm not exactly sure what you mean by (n|k) i think you are correct, but let me make it clearer
on second thought, this is a complicated equation to solve, and nothing too enlightening or complex is going on here.... if you are just using a calculator i would just keep increasing n by 1 until you get the probability to be >=.8
if you have a graphing calculator you should have a binomialcdf command that you can take advantage of... I am getting 16 as the answer
Do 1-binompdf(16,.1,0) and then do 1-binompdf(15,.1,0) you'll see that you need 16
ooor you could do
binomcdf(15,.1,1,15)=.79 and then binomcdf(16,.1,1,16)>.8
solving by hand seems pointless and unreasonable, but if you insist on doing it by hand I'm not sure I have any tips for you