Sequences and Series

Sequences as functions on the naturals — general terms, recursion, sigma notation — centred on the geometric progression: $a_n = ar^{n-1}$, the sum $S_n = \frac{a(r^n - 1)}{r - 1}$, geometric means, and the fundamental inequality $A.M. \ge G.M.$ Applications run from compound interest to the 7 + 77 + 777 trick, all core JEE material.