the math is fine
she just skipped a couple steps
S_m
= ax^0 + ax^1 + … + ax^(m–1)
= a(r^m – 1)/(r – 1)
S_2m
= a(r^2m – 1)/(r – 1) [formula]
= a((r^m)^2 – 1^2)(r^m + 1)/(r – 1) [a^(bc) = (a^b)^c]
= a(r^m – 1)(r^m + 1)/(r – 1) [difference of squares]
= a(r^m – 1)/(r – 1) * (r^m + 1)
= S_m *...