Triangular Numbers 1, 3, 6, 10, 15 Non-linear Pattern Rules -

Triangular numbers are "non-linear" because the difference between terms is not constant. Instead, the difference increases by 1 each time. 2. Recognize the quadratic nature

To find any term without listing the whole sequence, plug the position into the explicit formula. For example, to find the 100th triangular number: Triangular Numbers 1, 3, 6, 10, 15 Non-Linear Pattern Rules

T100=100(100+1)2=50×101=5050cap T sub 100 equals the fraction with numerator 100 open paren 100 plus 1 close paren and denominator 2 end-fraction equals 50 cross 101 equals 5050 ✅ Answer The rule for the triangular number sequence is Recognize the quadratic nature To find any term

Because the second difference is constant (always 1), the sequence is quadratic. This means the rule involves an n2n squared : Explicit Rule : 3. Visualize the geometry Visualize the geometry Each number represents the number

Each number represents the number of dots needed to form an equilateral triangle. To find the next number in the sequence, you simply add a new row of dots to the base of the previous triangle. 4. Apply the formula

The non-linear pattern of triangular numbers is defined by the rule that the -th triangular number ( Tncap T sub n ) is the sum of the first natural numbers, which simplifies to the quadratic formula Licensed by Google 1. Identify the constant difference