Three circles of radii `r_1, r_2 and r_3` are drawn concentric to each other. The radii `r_1` and `r_2` are such that the area of the circle with radius `r_1` is equal to the area between the circle of radius `r_2` and `r_1` . The area between the circle of radii `r_2` and `r_3` is equal to area between the circle of radii `r_2` and `r_r` . What is the value of `r_1 : r_2 : r_3` ?

Answer

To solve the problem step by step, we need to find the ratio \( r_1 : r_2 : r_3 \) based on the given conditions regarding the areas of the circles.

### Step 1: Understand the areas of the circles
The area \( A \) of a circle is given by the formula:
\[ A = \pi r^2 \]
For circles with radii \( r_1 \), \( r_2 \), and \( r_3 \), their respective areas will be:
- Area of circle with radius \( r_1 \): \( A_1 = \pi r_1^2 \)
- Area of circle with radius \( r_2 \): \( A_2 = \pi r_2^2 \)
- Area of circle with radius \( r_3 \): \( A_3 = \pi r_3^2 \)

### Step 2: Set up the first condition
The first condition states that the area of the circle with radius \( r_1 \) is equal to the area between the circles of radius \( r_2 \) and \( r_1 \):
\[ \pi r_1^2 = \pi (r_2^2 - r_1^2) \]
Cancelling \( \pi \) from both sides gives:
\[ r_1^2 = r_2^2 - r_1^2 \]
Rearranging this, we have:
\[ 2r_1^2 = r_2^2 \]
Thus,
\[ \frac{r_1^2}{r_2^2} = \frac{1}{2} \]
Taking the square root of both sides, we find:
\[ \frac{r_1}{r_2} = \frac{1}{\sqrt{2}} \]

### Step 3: Set up the second condition
The second condition states that the area between the circles of radius \( r_2 \) and \( r_3 \) is equal to the area between the circles of radius \( r_2 \) and \( r_1 \):
\[ \pi (r_3^2 - r_2^2) = \pi (r_2^2 - r_1^2) \]
Again, cancelling \( \pi \) gives:
\[ r_3^2 - r_2^2 = r_2^2 - r_1^2 \]
Rearranging this, we have:
\[ r_3^2 = 2r_2^2 - r_1^2 \]

### Step 4: Substitute \( r_1^2 \) in terms of \( r_2^2 \)
From the first condition, we know \( r_1^2 = \frac{1}{2} r_2^2 \). Substituting this into the equation for \( r_3^2 \):
\[ r_3^2 = 2r_2^2 - \frac{1}{2} r_2^2 \]
This simplifies to:
\[ r_3^2 = \frac{4}{2} r_2^2 - \frac{1}{2} r_2^2 = \frac{3}{2} r_2^2 \]

### Step 5: Find the ratio \( r_1 : r_2 : r_3 \)
Now we have:
- \( r_1^2 = \frac{1}{2} r_2^2 \)
- \( r_2^2 = r_2^2 \)
- \( r_3^2 = \frac{3}{2} r_2^2 \)

Taking square roots gives:
\[ r_1 = \frac{1}{\sqrt{2}} r_2, \quad r_2 = r_2, \quad r_3 = \sqrt{\frac{3}{2}} r_2 \]

Thus, the ratio \( r_1 : r_2 : r_3 \) can be expressed as:
\[ r_1 : r_2 : r_3 = \frac{1}{\sqrt{2}} : 1 : \sqrt{\frac{3}{2}} \]

### Step 6: Normalize the ratio
To express this in a more standard form, we can multiply through by \( \sqrt{2} \):
\[ r_1 : r_2 : r_3 = 1 : \sqrt{2} : \sqrt{3} \]

### Final Answer
The value of \( r_1 : r_2 : r_3 \) is:
\[ 1 : \sqrt{2} : \sqrt{3} \]