Three numbers a, b and c are in geometric progression. If 4a, 5b and 4c are in arithmetic progression and `a+b+c=70`, then the value of `|c-a|` is equal to

Options

A. 10
B. 20
C. 30
D. 40

Answer

To solve the problem step by step, we will follow the given conditions and derive the necessary equations.

### Step 1: Understand the conditions
We know that:
1. \( a, b, c \) are in geometric progression (GP).
2. \( 4a, 5b, 4c \) are in arithmetic progression (AP).
3. \( a + b + c = 70 \).

### Step 2: Use the GP condition
Since \( a, b, c \) are in GP, we can express the relationship between them:
\[ \frac{b}{a} = \frac{c}{b} \implies b^2 = ac \tag{1} \]

### Step 3: Use the AP condition
For the numbers \( 4a, 5b, 4c \) to be in AP, the difference between consecutive terms must be equal:
\[ 5b - 4a = 4c - 5b \]
Rearranging gives:
\[ 10b = 4c + 4a \implies 5b = 2c + 2a \tag{2} \]

### Step 4: Use the sum condition
From the sum condition:
\[ a + b + c = 70 \implies c = 70 - a - b \tag{3} \]

### Step 5: Substitute \( c \) in equation (2)
Substituting equation (3) into equation (2):
\[ 5b = 2(70 - a - b) + 2a \]
Expanding this gives:
\[ 5b = 140 - 2a - 2b + 2a \]
Simplifying results in:
\[ 5b + 2b = 140 \implies 7b = 140 \implies b = 20 \tag{4} \]

### Step 6: Find \( a + c \)
Using equation (3) with \( b = 20 \):
\[ a + 20 + c = 70 \implies a + c = 50 \tag{5} \]

### Step 7: Substitute \( b \) in equation (1)
Now substituting \( b = 20 \) into equation (1):
\[ 20^2 = ac \implies 400 = ac \tag{6} \]

### Step 8: Solve the system of equations
Now we have two equations:
1. \( a + c = 50 \) (from equation 5)
2. \( ac = 400 \) (from equation 6)

Let \( c = 50 - a \). Substitute into equation (6):
\[ a(50 - a) = 400 \]
Expanding gives:
\[ 50a - a^2 = 400 \implies a^2 - 50a + 400 = 0 \]

### Step 9: Solve the quadratic equation
Using the quadratic formula:
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{50 \pm \sqrt{2500 - 1600}}{2} = \frac{50 \pm \sqrt{900}}{2} = \frac{50 \pm 30}{2} \]
Thus, we have:
\[ a = \frac{80}{2} = 40 \quad \text{or} \quad a = \frac{20}{2} = 10 \]

### Step 10: Find corresponding \( c \) values
If \( a = 40 \):
\[ c = 50 - 40 = 10 \]
If \( a = 10 \):
\[ c = 50 - 10 = 40 \]

### Step 11: Calculate \( |c - a| \)
In both cases:
1. If \( a = 40 \) and \( c = 10 \), then \( |c - a| = |10 - 40| = 30 \).
2. If \( a = 10 \) and \( c = 40 \), then \( |c - a| = |40 - 10| = 30 \).

Thus, in both cases, we find:
\[ |c - a| = 30 \]

### Final Answer
The value of \( |c - a| \) is \( \boxed{30} \).