Meaning of Sequences and Geometric Progression

The Gujarat Board (GSHSEB) Std 11 Commerce Statistics syllabus builds on ordinary numbers arranged in a definite pattern, called a sequence. When the terms of a sequence follow a rule connecting each term to the next, the sequence is called a progression. This chapter studies one of the most useful progressions in business and finance: the Geometric Progression (GP).

A sequence of non-zero numbers $a_1, a_2, a_3, \ldots, a_n$ is called a Geometric Progression if the ratio of every term (except the first) to the term immediately preceding it is always the same constant. This constant is called the common ratio, denoted $r$:

The first term of a GP is usually denoted $a$ (or $a_1$).

Examples of GP:

• $3, 6, 12, 24, 48, \ldots$ — here $a = 3$, $r = 2$
• $2, -6, 18, -54, \ldots$ — here $a = 2$, $r = -3$
• $8, 4, 2, 1, \tfrac{1}{2}, \ldots$ — here $a = 8$, $r = \tfrac{1}{2}$

How to check whether a given sequence is a GP: divide each term by the term before it. If every such ratio comes out equal, the sequence is a GP; if even one ratio differs, it is not.

For example, $2, 4, 6, 8$ is not a GP because $\tfrac{4}{2} = 2$ but $\tfrac{6}{4} = 1.5$ — the ratios are not equal.

Geometric Progression is a standard, widely used topic across the Gujarat Secondary and Higher Secondary Education Board (GSHSEB) commerce Statistics syllabus, because it models real growth and decay patterns — compound interest, population growth, depreciation of assets, and annuities — all of which a commerce student meets again later in Accountancy and Economics.