Triangle numbers 1 - Reasoning/Problem Solving Maths Worksheets ... - Free Printable

Let's solve the problem step by step based on the image you've described.

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Task Overview:

The worksheet is about triangle numbers. It shows a pattern of dots arranged in a triangle, where each row adds one more dot than the previous row.

We are to:
1. Continue drawing the pattern.
2. Write down the sequence of triangle numbers.
3. Highlight two adjacent triangle numbers (e.g., 1 and 3).
4. Add those two numbers and observe what kind of number the result is.

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Step 1: Understanding Triangle Numbers

Triangle numbers are formed by adding consecutive natural numbers:

- 1 = 1
- 3 = 1 + 2
- 6 = 1 + 2 + 3
- 10 = 1 + 2 + 3 + 4
- 15 = 1 + 2 + 3 + 4 + 5
- etc.

So the sequence is:
1, 3, 6, 10, 15, 21, 28, ...

Each number represents the total number of dots in a triangular arrangement.

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Step 2: Drawing the Pattern

You're shown a triangle with dots:

```
•
• •
• • •
• • • •
```

This corresponds to the 4th triangle number: 1+2+3+4 = 10.

To continue the pattern, add another row with 5 dots:

```
•
• •
• • •
• • • •
• • • • •
```

This gives the 5th triangle number: 1+2+3+4+5 = 15.

And so on.

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Step 3: Sequence of Triangle Numbers

Write down the sequence:

1, 3, 6, 10, 15, 21, 28, ...

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Step 4: Highlight Two Adjacent Triangle Numbers

Example: Highlight 1 and 3.

Now add them:
1 + 3 = 4

What kind of number is 4?

✔ 4 is a square number! (Because 2 × 2 = 4)

Try another pair:
3 + 6 = 9 → 9 is also a square number (3×3)
6 + 10 = 16 → 16 is a square number (4×4)
10 + 15 = 25 → 25 is a square number (5×5)

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✔ Key Observation:

> The sum of two adjacent triangle numbers is always a square number!

This is a known mathematical pattern.

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Why Does This Happen?

Let’s look at the formula for the *n-th* triangle number:

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

Then the next triangle number is:

$$
T_{n+1} = \frac{(n+1)(n+2)}{2}
$$

Add them:

$$
T_n + T_{n+1} = \frac{n(n+1)}{2} + \frac{(n+1)(n+2)}{2}
= \frac{(n+1)}{2} [n + (n+2)] = \frac{(n+1)}{2} (2n+2)
= \frac{(n+1)}{2} \cdot 2(n+1) = (n+1)^2
$$

So the sum is $(n+1)^2$, which is a perfect square!

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✔ Final Answer:

- Sequence: 1, 3, 6, 10, 15, 21, ...
- Highlighted numbers: e.g., 1 and 3
- Sum: 1 + 3 = 4
- What kind of number? A square number (since $2^2 = 4$)

👉 Conclusion: The sum of any two adjacent triangle numbers is always a square number.

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Let me know if you'd like help continuing the drawing or writing more examples!