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:

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.