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### Topic: **Polygonal Numbers**
- **Objective:** Finding the formula for the nth term of triangular numbers.

### Triangular Numbers
- Triangular numbers are numbers that can form an equilateral triangle when represented as dots.
  
#### Sequence Example:
1 dot, 3 dots, 6 dots, 10 dots, 15 dots. These correspond to the sequence:
\[ 1, 3, 6, 10, 15 \]

### Common difference:
The difference between consecutive terms is increasing:
- \( 3 - 1 = 2 \)
- \( 6 - 3 = 3 \)
- \( 10 - 6 = 4 \)
- \( 15 - 10 = 5 \)

#### Second difference:
- The second difference (difference of differences) is constant:
  \[ 3 - 2 = 1 \]
  \[ 4 - 3 = 1 \]
  \[ 5 - 4 = 1 \]

Since the second difference is constant, this sequence is quadratic. Therefore, the formula for the nth term will be of the form:
\[
T_n = an^2 + bn + c
\]

### Step-by-step derivation of the nth-term formula:
1. **Use the first few terms to create a system of equations:**
   - \( T_1 = 1: \ a(1)^2 + b(1) + c = 1 \Rightarrow a + b + c = 1 \)
   - \( T_2 = 3: \ a(2)^2 + b(2) + c = 3 \Rightarrow 4a + 2b + c = 3 \)
   - \( T_3 = 6: \ a(3)^2 + b(3) + c = 6 \Rightarrow 9a + 3b + c = 6 \)

2. **Solve the system of equations:**
   - From the first equation: \( a + b + c = 1 \) (Eq. 1)
   - From the second equation: \( 4a + 2b + c = 3 \) (Eq. 2)
   - From the third equation: \( 9a + 3b + c = 6 \) (Eq. 3)

   Subtract Eq. 1 from Eq. 2:
   \[
   (4a + 2b + c) - (a + b + c) = 3 - 1
   \]
   \[
   3a + b = 2 \quad \text{(Eq. 4)}
   \]

   Subtract Eq. 2 from Eq. 3:
   \[
   (9a + 3b + c) - (4a + 2b + c) = 6 - 3
   \]
   \[
   5a + b = 3 \quad \text{(Eq. 5)}
   \]

3. **Solve for a and b:**
   Subtract Eq. 4 from Eq. 5:
   \[
   (5a + b) - (3a + b) = 3 - 2
   \]
   \[
   2a = 1 \quad \Rightarrow \ a = \frac{1}{2}
   \]

   Substitute \( a = \frac{1}{2} \) into Eq. 4:
   \[
   3\left(\frac{

Find the nth term formula for triangular numbers.

Learn how to find the nth term formula for triangular numbers using quadratic sequences. This problem covers arithmetic sequences, systems of equations, and solving for a, b, and c. Suitable for Grades 8-10.

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