Sequences and Series — The Basic Idea

A sequence is an ordered list of numbers written according to some definite rule — for example $2, 4, 6, 8, \ldots$ or $1, 3, 9, 27, \ldots$. Each number in the list is a term, and the terms are written $t_1, t_2, t_3, \ldots$ (some books use $a_1, a_2, \ldots$), where $t_n$ denotes the $n$-th term — the term in position $n$.

When the terms of a sequence are joined by plus signs and added together, the result is called a series. Thus if the sequence is $t_1, t_2, t_3, \ldots, t_n$, the corresponding series is

and its value (the total of the first $n$ terms) is denoted $S_n$.

This chapter of the Maharashtra Std XI (FYJC) commerce Mathematics and Statistics course studies the three most useful patterned sequences — Arithmetic Progression (AP), Geometric Progression (GP) and Harmonic Progression (HP) — together with their means (AM, GM, HM) and the standard sums of powers of natural numbers. These are the same sequences-and-series principles set out in the national NCERT/CBSE mathematics curriculum, and they underpin later commercial-mathematics work on instalments, depreciation, annuities and growth. For each we build two tools: a formula for any single term $t_n$, and a formula for the running total $S_n$.