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To solve for the **hexagonal numbers** using the provided example and notes, let's first break down the general formula used for hexagonal numbers.

### General Formula for Hexagonal Numbers:
The \(n\)-th hexagonal number can be given by the formula:
\[
H_n = 2n^2 - n
\]
Where:
- \(n\) is the position of the number in the sequence (starting from \(n = 1\)).

### Example from the notes:
The handwritten notes seem to show a quadratic expression involving the terms \( a, b, \text{and } c \), and there is some substitution happening, likely attempting to derive a form for the hexagonal number sequence.

Here’s a typical breakdown based on what’s shown:

1. **Quadratic equation form**: The general form used to calculate terms appears to be:
\[
U_n = a n^2 + b n + c
\]
Where \(a\), \(b\), and \(c\) are constants that need to be determined using known values from the sequence.

2. **Use Known Values**: 
   Based on the hexagonal numbers:
   - \( U_1 = 1 \)
   - \( U_2 = 6 \)
   - \( U_3 = 15 \)

   These values are substituted into the quadratic equation to solve for \(a\), \(b\), and \(c\).

### Solving for Constants \(a\), \(b\), and \(c\):
From the notes:
\[
U_1 = a(1)^2 + b(1) + c = 1
\]
\[
U_2 = a(2)^2 + b(2) + c = 6
\]
\[
U_3 = a(3)^2 + b(3) + c = 15
\]
Simplifying:
- \(a + b + c = 1\)
- \(4a + 2b + c = 6\)
- \(9a + 3b + c = 15\)

By solving these equations, you get the values for \(a\), \(b\), and \(c\). The handwritten notes show that:
- \(a = \frac{3}{2}\)
- \(b = -\frac{1}{2}\)
- \(c = 0\)

Thus, the quadratic equation becomes:
\[


Solve for hexagonal numbers using this example

Discover how to calculate hexagonal numbers using a quadratic equation. Learn to solve systems of equations to find constants for the hexagonal sequence using known values like 1, 6, and 15. This tutorial is perfect for Grades 9-12 students studying algebra and number theory.

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