State True or False on a dot pattern of numbers:
Number 9 can be expressed as a triangle.

Answer

To determine whether the number 9 can be expressed as a triangle using a dot pattern, we can follow these steps:

### Step 1: Understand Triangle Numbers
Triangle numbers are formed by arranging dots in the shape of an equilateral triangle. The nth triangle number can be calculated using the formula:
\[ T_n = \frac{n(n + 1)}{2} \]
where \( n \) is the number of rows.

### Step 2: Calculate Triangle Numbers
Let's calculate the first few triangle numbers to see if 9 is one of them.

- For \( n = 1 \):
\[ T_1 = \frac{1(1 + 1)}{2} = \frac{1 \times 2}{2} = 1 \]

- For \( n = 2 \):
\[ T_2 = \frac{2(2 + 1)}{2} = \frac{2 \times 3}{2} = 3 \]

- For \( n = 3 \):
\[ T_3 = \frac{3(3 + 1)}{2} = \frac{3 \times 4}{2} = 6 \]

- For \( n = 4 \):
\[ T_4 = \frac{4(4 + 1)}{2} = \frac{4 \times 5}{2} = 10 \]

### Step 3: Analyze the Results
From our calculations, we have the following triangle numbers:
- \( T_1 = 1 \)
- \( T_2 = 3 \)
- \( T_3 = 6 \)
- \( T_4 = 10 \)

We can see that 9 is not listed among these triangle numbers.

### Step 4: Conclusion
Since 9 does not appear in the list of triangle numbers, we conclude that **the statement is False**. Number 9 cannot be expressed as a triangle.

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