If x+1, 4x+1, and 8x+1 are in geometric progression, then what is the non-trivial value of x ?

Options

A. `-1`
B. `1`
C. `1/8`
D. `1/4`

Answer

To solve the problem, we need to determine the non-trivial value of \( x \) such that \( x + 1 \), \( 4x + 1 \), and \( 8x + 1 \) are in geometric progression (GP).

### Step-by-step Solution:

1. **Understanding Geometric Progression**:
For three numbers \( a \), \( b \), and \( c \) to be in GP, the ratio of the first two terms must equal the ratio of the last two terms. This can be expressed mathematically as:
\[ \frac{b}{a} = \frac{c}{b} \]

2. **Setting up the equation**:
Here, let \( a = x + 1 \), \( b = 4x + 1 \), and \( c = 8x + 1 \). Therefore, we can set up the equation:
\[ \frac{4x + 1}{x + 1} = \frac{8x + 1}{4x + 1} \]

3. **Cross-multiplying**:
Cross-multiply to eliminate the fractions:
\[ (4x + 1)^2 = (x + 1)(8x + 1) \]

4. **Expanding both sides**:
- Left-hand side:
\[ (4x + 1)^2 = 16x^2 + 8x + 1 \]
- Right-hand side:
\[ (x + 1)(8x + 1) = 8x^2 + x + 8x + 1 = 8x^2 + 9x + 1 \]

5. **Setting the equation**:
Now we have:
\[ 16x^2 + 8x + 1 = 8x^2 + 9x + 1 \]

6. **Rearranging the equation**:
Move all terms to one side:
\[ 16x^2 + 8x + 1 - 8x^2 - 9x - 1 = 0 \]
This simplifies to:
\[ 8x^2 - x = 0 \]

7. **Factoring the equation**:
Factor out \( x \):
\[ x(8x - 1) = 0 \]

8. **Finding the values of \( x \)**:
This gives us two solutions:
\[ x = 0 \quad \text{or} \quad 8x - 1 = 0 \Rightarrow x = \frac{1}{8} \]

9. **Identifying the non-trivial solution**:
The trivial solution is \( x = 0 \). Therefore, the non-trivial solution is:
\[ x = \frac{1}{8} \]

### Final Answer:
The non-trivial value of \( x \) is \( \frac{1}{8} \).