Find the value of x for which x +9,x-6, 4 are the first three terms of a geometrical progression and calculate the fourth term of progression in this case

Answer

To find the value of \( x \) for which \( x + 9, x - 6, 4 \) are the first three terms of a geometric progression (GP), we can follow these steps:

### Step 1: Set up the relationship for a GP
In a geometric progression, the ratio of consecutive terms is constant. Therefore, we can write:
\[ \frac{x - 6}{x + 9} = \frac{4}{x - 6} \]

### Step 2: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ (x - 6)^2 = 4(x + 9) \]

### Step 3: Expand both sides
Expanding both sides, we have:
\[ x^2 - 12x + 36 = 4x + 36 \]

### Step 4: Rearrange the equation
Now, rearranging the equation to one side:
\[ x^2 - 12x - 4x + 36 - 36 = 0 \]
This simplifies to:
\[ x^2 - 16x = 0 \]

### Step 5: Factor the quadratic equation
Factoring out \( x \):
\[ x(x - 16) = 0 \]

### Step 6: Solve for \( x \)
Setting each factor to zero gives us:
\[ x = 0 \quad \text{or} \quad x = 16 \]

### Step 7: Determine the valid value for \( x \)
Since \( x + 9 \) and \( x - 6 \) must be positive for them to be valid terms in a GP, we can check:
- For \( x = 0 \): \( x + 9 = 9 \) (valid), \( x - 6 = -6 \) (invalid)
- For \( x = 16 \): \( x + 9 = 25 \) (valid), \( x - 6 = 10 \) (valid)

Thus, the valid value of \( x \) is \( 16 \).

### Step 8: Calculate the fourth term of the GP
Now, we need to find the fourth term. We first calculate \( a \) and \( r \):
- \( a = x + 9 = 16 + 9 = 25 \)
- \( r = \frac{4}{x - 6} = \frac{4}{16 - 6} = \frac{4}{10} = 0.4 \)

The fourth term \( a_4 \) can be calculated using the formula:
\[ a_n = ar^{n-1} \]
For \( n = 4 \):
\[ a_4 = 25 \cdot (0.4)^{3} \]
Calculating \( (0.4)^{3} = 0.064 \):
\[ a_4 = 25 \cdot 0.064 = 1.6 \]

### Final Answer
The value of \( x \) is \( 16 \) and the fourth term of the progression is \( 1.6 \).

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