Use patterns to complete the table below.

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Part 1

**Figurate Number** 

**1st**

**2nd**

**3rd**

**4th**

**5th**

**6th**

**7th**

**8th**

Triangular

1

3

6

10

15

21

enter your response here

enter your response hereUse the formula S 

​=StartFraction n left parenthesis n plus 1 right parenthesis Over 2 EndFractionn(n + 1)2 

to find the sum of 1​ + 2​ + 3​ + ...​ + 

930930.

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Part 1

1​ + 2​ + 3​ + ...​ + 930

​=

### Completing the Triangular Number Sequence:

The sequence of triangular numbers is generated by the formula:

\[
T_n = \frac{n(n + 1)}{2}
\]

Where \( T_n \) is the \( n \)-th triangular number.

- **1st Triangular Number**: \( T_1 = \frac{1(1 + 1)}{2} = 1 \)
- **2nd Triangular Number**: \( T_2 = \frac{2(2 + 1)}{2} = 3 \)
- **3rd Triangular Number**: \( T_3 = \frac{3(3 + 1)}{2} = 6 \)
- **4th Triangular Number**: \( T_4 = \frac{4(4 + 1)}{2} = 10 \)
- **5th Triangular Number**: \( T_5 = \frac{5(5 + 1)}{2} = 15 \)
- **6th Triangular Number**: \( T_6 = \frac{6(6 + 1)}{2} = 21 \)

Let's calculate the 7th and 8th triangular numbers:

- **7th Triangular Number**: \( T_7 = \frac{7(7 + 1)}{2} = \frac{7 \times 8}{2} = 28 \)
- **8th Triangular Number**: \( T_8 = \frac{8(8 + 1)}{2} = \frac{8 \times 9}{2} = 36 \)

So, the table should be completed as:

| **Triangular** | **1st** | **2nd** | **3rd** | **4th** | **5th** | **6th** | **7th** | **8th** |
|----------------|---------|---------|---------|---------|---------|---------|---------|---------|
| **Number**     | 1       | 3       | 6       | 10      | 15      | 21      | 28      | 36      |

### Finding the Sum of 1 + 2 + 3 + ... + 930:

To find the sum of the first 930 natural numbers, we use the formula:

\[
S = \frac{n(n + 1)}{2}
\]

Where \( n = 930 \):

\[
S = \frac{930 \times 931}{2} = \frac{865830}{2} = 432915
\]

So, the sum of the numbers from 1 to 930 is **432,915**.

Would you like more details or have any questions?

### Related Questions:

1. How do you derive the formula for triangular numbers?
2. What are square numbers, and how are they related to triangular numbers?
3. Can you use the triangular number formula to find the 100th triangular number?
4. How do triangular numbers relate to the sum of consecutive integers?
5. What other sequences are there in figurate numbers, such as pentagonal or hexagonal?

**Tip:** For quick calculations, memorizing the formulas for special sequences like triangular numbers can save time on tests or assignments.

Learn how to complete the triangular number sequence and find the sum of the first 930 natural numbers using formulas. This problem covers triangular numbers and their sequence, along with calculating summations of natural numbers up to 930. Suitable for students in Grades 6-8.

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