Stack the rows

The idea

Each triangular number is the sum of the natural numbers up to its position: 1, 1 + 2, 1 + 2 + 3, and so on.

Dots in a triangle

Stack dots in a triangle: 1 dot, then 3, then 6, then 10, then 15 … These are the triangular numbers: 1, 3, 6, 10, 15, 21, …

The 3rd triangle has 6 dots and the 4th has 10. How many dots were added to the 3rd to make the 4th?

• 3
• 4
• 5

Build the triangle

Add one row at a time. Watch the sum grow under the picture.

Sums of natural numbers

Each triangular number is the sum of the natural numbers up to its position: 1, 1 + 2, 1 + 2 + 3, and so on.

Every new triangle adds one more row than the last, so the differences are 2, 3, 4, 5, 6, …

• Position | Triangular number
• 1st | 1 = 1
• 2nd | 3 = 1 + 2
• 3rd | 6 = 1 + 2 + 3
• 4th | 10 = 1 + 2 + 3 + 4
• 5th | 15 = 1 + 2 + 3 + 4 + 5

Notes

Each triangular number is the sum of the natural numbers up to its position: 1, 1 + 2, 1 + 2 + 3, and so on.

Check yourself

What is the 6th triangular number?

Answer: 21

1 + 2 + 3 + 4 + 5 + 6 = 21. Or: the 5th triangular number is 15, and the 6th adds a row of 6 dots, so 15 + 6 = 21.

Which sum gives the 4th triangular number?

The 7th triangular number is 28. The 8th adds a new row. How many dots are in that new row?

Answer: 8

The 8th triangular number is 1 + 2 + … + 7 + 8. The new row has 8 dots (28 + 8 = 36).

The 9th triangular number is 45. What is the 10th triangular number?

Answer: 55

45 + 10 = 55. The 10th triangular number is 1 + 2 + … + 10 = 55.

• 1 + 2 + 3 + 4 — correct. Yes. The 4th triangular number is the sum of the natural numbers up to 4: 1 + 2 + 3 + 4 = 10.
• 1 + 2 + 3 + 4 + 5. This goes up to 5, so it is the 5th triangular number (15).
• 1 + 3 + 5 + 7. These are odd numbers. Triangular numbers add the natural numbers 1, 2, 3, 4 in a row.