Patterns in Mathematics - Patterns in Numbers

A number pattern is a sequence of numbers that follows a specific rule. For example, in the sequence $5, 10, 15, 20, \dots$, the rule is to add $5$ to the previous term.

Patterns in Addition: Adding numbers like $9, 99, 999$ can be simplified using the logic $n + 9 = n + 10 - 1$ and $n + 99 = n + 100 - 1$.

Patterns in Multiplication: Multiplying by $5, 25, 125$ can be simplified by using powers of $10$ and $2$. For example, $a \times 5 = a \times \frac{10}{2}$ and $a \times 25 = a \times \frac{100}{4}$.

Triangular Numbers: These are numbers that can be arranged in the shape of an equilateral triangle. The sequence is $1, 3, 6, 10, 15, \dots$. The $n^{th}$ triangular number is given by the sum of first $n$ natural numbers.

Square Numbers: These are numbers that can be arranged in a square shape. The sequence is $1, 4, 9, 16, 25, \dots$. The $n^{th}$ square number is $n \times n$ or $n^2$.

Even and Odd Patterns: The sum of two even numbers is always even ($2 + 4 = 6$). The sum of two odd numbers is always even ($3 + 5 = 8$). The sum of an even and an odd number is always odd ($2 + 3 = 5$).