Number Patterns

Triangular Patterns

Consider a pattern of circles arranged in triangles or squares. If we examine the number of items in each step, we can often find a quadratic formula ($an^2 + bn + c$).

Example:

• Step 1: 3 items
• Step 2: 8 items
• Step 3: 15 items

Let’s look at the structure.

• $3 = 1 \times 3$
• $8 = 2 \times 4$
• $15 = 3 \times 5$

The pattern is $k \times (k+2)$. Expression: $k^2 + 2k$.

Sum of Squares Pattern

$2(a^2 + b^2) = (a+b)^2 + (a-b)^2$

This identity connects the sum of squares to the square of sum and square of difference.

Verification: RHS $= (a^2 + 2ab + b^2) + (a^2 - 2ab + b^2)$ $= 2a^2 + 2b^2$ $= 2(a^2 + b^2)$ = LHS.