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 information and derive the required values.

### Step 1: Understand the relationship between a, b, and c
Since a, b, and c are in geometric progression (G.P.), we can express them in terms of a common ratio \( r \):
- Let \( a = a \)
- \( b = ar \)
- \( c = ar^2 \)

### Step 2: Use the sum condition
We are given that:
\[ a + b + c = 70 \]
Substituting the expressions for b and c:
\[ a + ar + ar^2 = 70 \]
Factoring out \( a \):
\[ a(1 + r + r^2) = 70 \]

### Step 3: Use the arithmetic progression condition
We are also given that \( 4a, 5b, 4c \) are in arithmetic progression (A.P.). For three numbers to be in A.P., the middle term must be the average of the other two:
\[ 2 \cdot 5b = 4a + 4c \]
Substituting \( b \) and \( c \):
\[ 10(ar) = 4a + 4(ar^2) \]
This simplifies to:
\[ 10ar = 4a + 4ar^2 \]
Rearranging gives:
\[ 4ar^2 - 10ar + 4a = 0 \]
Factoring out \( 2a \):
\[ 2a(2r^2 - 5r + 2) = 0 \]
Since \( a \neq 0 \), we have:
\[ 2r^2 - 5r + 2 = 0 \]

### Step 4: Solve the quadratic equation
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2, b = -5, c = 2 \):
\[ r = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \]
\[ r = \frac{5 \pm \sqrt{25 - 16}}{4} \]
\[ r = \frac{5 \pm 3}{4} \]
Thus, we have two possible values for \( r \):
1. \( r = 2 \)
2. \( r = \frac{1}{2} \)

### Step 5: Find values of a, b, and c for each case
**Case 1: \( r = 2 \)**
Substituting \( r = 2 \) back into the sum equation:
\[ a(1 + 2 + 4) = 70 \implies a \cdot 7 = 70 \implies a = 10 \]
Then:
- \( b = ar = 10 \cdot 2 = 20 \)
- \( c = ar^2 = 10 \cdot 4 = 40 \)

**Case 2: \( r = \frac{1}{2} \)**
Substituting \( r = \frac{1}{2} \):
\[ a(1 + \frac{1}{2} + \frac{1}{4}) = 70 \implies a \cdot \frac{7}{4} = 70 \implies a = 40 \]
Then:
- \( b = ar = 40 \cdot \frac{1}{2} = 20 \)
- \( c = ar^2 = 40 \cdot \frac{1}{4} = 10 \)

### Step 6: Calculate \( |c - a| \)
In both cases:
- For \( r = 2 \): \( |c - a| = |40 - 10| = 30 \)
- For \( r = \frac{1}{2} \): \( |c - a| = |10 - 40| = 30 \)

### Conclusion
In both cases, we find that:
\[ |c - a| = 30 \]

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