Order of Summation

Clear[a]

red[m_]:=Sum[a[2k-1]-a[2k],{k,1,3m}]
red[1]
red[2]

blue[m_]:=Sum[a[4k-3]+a[4k-1]-a[2k],{k,1,2m}]
blue[1]
blue[2]

a[k_]=1/k^2
red[1];
(a[1]-a[2])+(a[3]-a[4])+(a[5]-a[6]);
blue[1];
(a[1]+a[3]-a[2])+(a[5]+a[7]-a[4]);

z=50;
pl1=ListPlot[Table[{6m,red[m]},{m,1,z,Ceiling[z/100]}],PlotStyle->{Thick,Red}];
pl2=ListPlot[Table[{6m,blue[m]},{m,1,z,Ceiling[z/100]}],PlotStyle->{Thick,Blue}];
Show[{pl1,pl2},PlotRange->All]

Another Example

For a[k]=(1/(Sqrt[k])):