S3m = 2 10

arithmetical progression

Sm = a 1 + a2 + a3 + …… + am

s2m-Sm = a(m+ 1)+a(m+2)+a(m+3)+……+a2m

s3m-S2m = a(2m+ 1)+a(2m+2)+a(2m+3)+……+a3m

Look at each column above. They are related. A 1, a(m+ 1) and a(2m+ 1) are just arithmetic progression with tolerance md, while a2, a(m+2) and a(2m+2) are just arithmetic progression with tolerance MD.

That is, sm+DM 2 = s2m-sm.

S2m-Sm +dm^2 = S3m-S2m

So Sm S2m-Sm S3m-S2m constitutes a arithmetic progression.