Show whether:
Number 28 can be expressed as a triangle of dots.

Answer

To determine whether the number 28 can be expressed as a triangle of dots, we need to understand how triangular numbers are formed. Triangular numbers are formed by arranging dots in the shape of an equilateral triangle. The nth triangular number can be calculated using the formula:

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

where \( n \) is the number of dots along one side of the triangle.

### Step-by-Step Solution:

1. **Identify the Formula for Triangular Numbers**:
The formula for the nth triangular number is given by:
\[ T_n = \frac{n(n + 1)}{2} \]

2. **Set Up the Equation for 28**:
We need to find \( n \) such that:
\[ T_n = 28 \]
This means we set up the equation:
\[ \frac{n(n + 1)}{2} = 28 \]

3. **Multiply Both Sides by 2**:
To eliminate the fraction, multiply both sides by 2:
\[ n(n + 1) = 56 \]

4. **Rearrange the Equation**:
Rearranging gives us a standard quadratic equation:
\[ n^2 + n - 56 = 0 \]

5. **Factor the Quadratic Equation**:
Now we need to factor the quadratic equation. We look for two numbers that multiply to -56 and add to 1. The numbers 8 and -7 work:
\[ (n + 8)(n - 7) = 0 \]

6. **Solve for n**:
Setting each factor to zero gives us:
\[ n + 8 = 0 \quad \text{or} \quad n - 7 = 0 \]
This results in:
\[ n = -8 \quad \text{or} \quad n = 7 \]
Since \( n \) must be a positive integer, we take \( n = 7 \).

7. **Verify the Result**:
Now we can verify by substituting \( n = 7 \) back into the triangular number formula:
\[ T_7 = \frac{7(7 + 1)}{2} = \frac{7 \times 8}{2} = \frac{56}{2} = 28 \]

### Conclusion:
Since we have found \( n = 7 \) and verified that \( T_7 = 28 \), we conclude that the number 28 can indeed be expressed as a triangle of dots.