The concepts of arithmetic and geometric progressions (series) appear in Vedic texts as early as 2000 BCE. These sequences were used in various contexts, including ritualistic calculations, altar constructions, and philosophical discussions.

1. Taittiriya Samhita (part of the Krishna Yajurveda)

Mentions progressions of numbers in ritualistic contexts.

Discusses successive offerings in yajñas (Vedic sacrifices), where offerings increase in a fixed pattern (an arithmetic series).

Example: If the first offering is of size aa and each subsequent offering increases by a fixed amount dd, the total offerings form an arithmetic progression (AP).

2. Pañcaviṃśa Brāhmaṇa (Sāmaveda Brahmana, c. 2000 BCE)

Contains geometric sequences related to multiplicative growth in rituals and cosmology.

Describes sequences where each term is a multiple of the previous term, forming a geometric progression (GP).

Example: The doubling of sacrificial materials follows a GP of ratio r=2r = 2 (1, 2, 4, 8, 16, …).

A sequence where the difference between consecutive terms is constant:

a,a+d,a+2d,a+3d,…a, a + d, a + 2d, a + 3d, \dots

Used in Vedic rituals for stepwise increases in offerings and mantra repetitions.

A sequence where each term is multiplied by a fixed ratio:

a,ar,ar2,ar3,…a, ar, ar^2, ar^3, \dots

Used in cosmological theories, multiplicative growth in yajñas, and construction of altars.

Later Indian mathematicians like Aryabhata (5th century CE) and Bhaskara II (12th century CE) formalized and expanded these concepts.

The principles of AP and GP were later used in astronomy, algebra, and financial calculations.

These ideas predate similar concepts found in Greek and Babylonian mathematics.

The early understanding of series in Vedic texts influenced later Indian and global mathematics.

It shows India’s deep mathematical heritage, stretching back over 4,000 years.