What should be added to each of the numbers `13`, `17 `and `22` so that the resulting numbers are in continued proportion ?

Answer

To solve the problem of what should be added to each of the numbers 13, 17, and 22 so that the resulting numbers are in continued proportion, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variable:**
Let \( x \) be the number that needs to be added to each of the numbers 13, 17, and 22.

2. **Formulate the New Numbers:**
The new numbers after adding \( x \) will be:
- First number: \( 13 + x \)
- Second number: \( 17 + x \)
- Third number: \( 22 + x \)

3. **Set Up the Proportion:**
For the numbers to be in continued proportion, the following relationship must hold:
\[ \frac{13 + x}{17 + x} = \frac{17 + x}{22 + x} \]

4. **Cross-Multiply:**
Cross-multiplying gives us:
\[ (13 + x)(22 + x) = (17 + x)(17 + x) \]

5. **Expand Both Sides:**
Expanding both sides:
- Left side:
\[ 13 \cdot 22 + 13x + 22x + x^2 = 286 + 35x + x^2 \]
- Right side:
\[ (17 + x)(17 + x) = 289 + 34x + x^2 \]

6. **Set the Equation:**
Now we have:
\[ 286 + 35x + x^2 = 289 + 34x + x^2 \]

7. **Simplify the Equation:**
We can eliminate \( x^2 \) from both sides:
\[ 286 + 35x = 289 + 34x \]

8. **Rearrange the Equation:**
Rearranging gives us:
\[ 35x - 34x = 289 - 286 \]
\[ x = 3 \]

9. **Conclusion:**
Therefore, the number that should be added to each of the numbers 13, 17, and 22 is \( \boxed{3} \).