The first three terms of a G.P. are x x +3, x+ 9. Find the value of x and the sum of first eight terms.

Answer

To solve the problem step by step, we will follow the properties of a geometric progression (G.P.) and use the given terms to find the value of \( x \) and the sum of the first eight terms.

### Step 1: Set up the equation using the property of G.P.
Given the first three terms of the G.P. are \( x \), \( x + 3 \), and \( x + 9 \), we can use the property that for three terms \( a \), \( b \), and \( c \) in G.P., the relationship \( a \cdot c = b^2 \) holds true.

Here, we can set up the equation:
\[ x(x + 9) = (x + 3)^2 \]

### Step 2: Expand both sides of the equation
Expanding both sides, we have:
\[ x^2 + 9x = (x + 3)(x + 3) = x^2 + 6x + 9 \]

### Step 3: Rearrange the equation
Now, we can rearrange the equation by moving all terms to one side:
\[ x^2 + 9x - (x^2 + 6x + 9) = 0 \]
This simplifies to:
\[ 9x - 6x - 9 = 0 \]
\[ 3x - 9 = 0 \]

### Step 4: Solve for \( x \)
Now, we can solve for \( x \):
\[ 3x = 9 \implies x = 3 \]

### Step 5: Find the terms of the G.P.
Now that we have \( x = 3 \), we can find the first three terms of the G.P.:
- First term: \( x = 3 \)
- Second term: \( x + 3 = 3 + 3 = 6 \)
- Third term: \( x + 9 = 3 + 9 = 12 \)

### Step 6: Find the common ratio \( r \)
The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{6}{3} = 2 \]

### Step 7: Calculate the sum of the first eight terms
The sum of the first \( n \) terms of a G.P. can be calculated using the formula:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

Substituting the known values:
- \( a = 3 \)
- \( r = 2 \)
- \( n = 8 \)

We can calculate \( S_8 \):
\[ S_8 = 3 \frac{2^8 - 1}{2 - 1} = 3 \frac{256 - 1}{1} = 3 \times 255 = 765 \]

### Final Answers
- The value of \( x \) is \( 3 \).
- The sum of the first eight terms is \( 765 \).