Triangular Numbers worksheet - Free Printable

Let's solve this step by step.

Understanding Triangular Numbers

Triangular numbers are formed by adding consecutive natural numbers starting from 1. Each triangular number represents the total number of dots that can be arranged in an equilateral triangle.

The pattern is:

- 1st triangular number: $ 1 = 1 $
- 2nd triangular number: $ 1 + 2 = 3 $
- 3rd triangular number: $ 1 + 2 + 3 = 6 $
- 4th triangular number: $ 1 + 2 + 3 + 4 = 10 $
- 5th triangular number: $ 1 + 2 + 3 + 4 + 5 = 15 $
- 6th triangular number: $ 1 + 2 + 3 + 4 + 5 + 6 = 21 $

So, the sequence of triangular numbers is:
$$
1,\ 3,\ 6,\ 10,\ 15,\ 21,\ \ldots
$$

We can see that each number is the sum of the first $ n $ natural numbers.

There is a formula for the $ n $th triangular number:
$$
T_n = \frac{n(n+1)}{2}
$$

---

Given Chart

The chart shows:

| 1 | 3 | 6 |
|---|---|---|
|10 |15 |21 |
| | | |
| | | |

We need to fill in the next three triangular numbers after 21.

Let’s compute them:

#### 7th triangular number:
$$
T_7 = 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28
$$
Or using the formula:
$$
T_7 = \frac{7(7+1)}{2} = \frac{7 \times 8}{2} = 28
$$

#### 8th triangular number:
$$
T_8 = \frac{8 \times 9}{2} = 36
$$

#### 9th triangular number:
$$
T_9 = \frac{9 \times 10}{2} = 45
$$

---

Filling the Chart

Now we complete the chart:

| 1 | 3 | 6 |
|---|---|---|
|10 |15 |21 |
|28 |36 |45 |
| | | |

But wait — the chart has three rows with three columns, and only the first two rows are filled. The third row is empty, so we should fill the next three triangular numbers (after 21), which are:

- 28 (7th)
- 36 (8th)
- 45 (9th)

So, the completed chart would be:

| 1 | 3 | 6 |
|---|---|---|
|10 |15 |21 |
|28 |36 |45 |

> ✔ So, the next triangular number after 21 is 28, then 36, then 45.

---

Answer:

The next triangular number that comes after 21 is 28.

If you're filling in the blank row in the chart, the correct entries are:

- First blank row: 28, 36, 45

✔ Final Answer: 28 (as the immediate next number)
And the full table becomes:

| 1 | 3 | 6 |
|---|---|---|
|10 |15 |21 |
|28 |36 |45 |

Let me know if you'd like the next few too!