You're replacing the contents of the textarea every time. If you want multiple lines, you'll need to concatenate them together.
EDIT:
Something like this. I can't run this on the site, but you can test it.
% placeholders
n = 1;
a0 = 1;
a = rand(1,n);
b = rand(1,n);
j1 = 1;
rsq1 = 1;
RMSE1 = 1;
% dummy figure
fig = uifigure;
app.TextArea = uitextarea(fig);
app.TextArea.Position(3:4) = [200 200];
alltext = sprintf('Fourier Number of Terms:%d\n\nEquation:\ny=a0',n);
for i=1:n
txt = '+a%d cos(%dx)+b%d sin(%dx)';
alltext = [alltext sprintf(txt,i,i,i,i)];
end
alltext = [alltext sprintf('\nCoefficients:\na0=%.4f',a0)];
for i=1:n
alltext = [alltext sprintf('\na%d=%.4f \nb%d=%.4f',i,a(i),i,b(i))];
end
alltext = [alltext sprintf('\n\nJ=%.4f \n\nR^2=%.4f \n\nRMSE=%.4f',j1,rsq1,RMSE1)];
app.TextArea.Value = alltext; % write once
