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Number Sequences

Among the most basic patterns in mathematics are number sequences. A sequence is simply a list of numbers that follow a specific rule.

The most fundamental sequence is the set of Whole Numbers: $0, 1, 2, 3, 4, \dots$

Key Number Sequences

Table 1 lists some of the most important sequences studied in mathematics.

Sequence Name	Terms	Pattern / Rule
All 1’s	$1, 1, 1, 1, 1, \dots$	Constant value.
Counting Numbers	$1, 2, 3, 4, 5, 6, 7, \dots$	Increasing by 1.
Odd Numbers	$1, 3, 5, 7, 9, 11, 13, \dots$	Numbers not divisible by 2.
Even Numbers	$2, 4, 6, 8, 10, 12, 14, \dots$	Numbers divisible by 2.
Triangular Numbers	$1, 3, 6, 10, 15, 21, 28, \dots$	Numbers that can form a triangle.
Square Numbers	$1, 4, 9, 16, 25, 36, 49, \dots$	Numbers that can form a square ( $n^2$ ).
Cubes	$1, 8, 27, 64, 125, 216, \dots$	Numbers formed by $n^3$ .
Virahānka Numbers	$1, 2, 3, 5, 8, 13, 21, \dots$	Each number is the sum of the previous two (often called Fibonacci).
Powers of 2	$1, 2, 4, 8, 16, 32, 64, \dots$	Doubling the previous number.
Powers of 3	$1, 3, 9, 27, 81, 243, \dots$	Tripling the previous number.

Tip

Did You Know? The Virahānka numbers (1, 2, 3, 5, 8…) are widely known as Fibonacci numbers in the West, but they were described earlier by Indian mathematician Virahānka in the context of analyzing poetic meters.

Recognizing Rules

To find the rule of a sequence, look at the difference between consecutive terms or the ratio between them.

Linear Growth: If the difference is constant (e.g., Counting Numbers $+1$ , Odd Numbers $+2$ ).
Exponential Growth: If the ratio is constant (e.g., Powers of 2 $\times 2$ ).
Geometric Growth: Sequences like Square numbers grow based on the size of the shape they represent.

\text{Square Number (n)} = n \times n = n^2

\text{Cube Number (n)} = n \times n \times n = n^3

Patterns in Numbers

Number Sequences

Key Number Sequences

Recognizing Rules