Find the sum of 8 terms of the G.P. : `3+6+12+24+ . . . . .. . . . . .`

Answer

To find the sum of the first 8 terms of the geometric progression (G.P.) given by the series \(3, 6, 12, 24, \ldots\), we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \(a\) of the G.P. is the first number in the series, which is:
\[ a = 3 \]
Next, we need to find the common ratio \(r\). The common ratio is found by dividing the second term by the first term:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{6}{3} = 2 \]

### Step 2: Determine the number of terms
We are asked to find the sum of the first 8 terms, so:
\[ n = 8 \]

### Step 3: Use the formula for the sum of the first \(n\) terms of a G.P.
The formula for the sum of the first \(n\) terms of a G.P. is given by:
\[ S_n = a \frac{r^n - 1}{r - 1} \]
Substituting the values we have:
- \(a = 3\)
- \(r = 2\)
- \(n = 8\)

We can write:
\[ S_8 = 3 \frac{2^8 - 1}{2 - 1} \]

### Step 4: Calculate \(2^8\)
First, calculate \(2^8\):
\[ 2^8 = 256 \]

### Step 5: Substitute back into the sum formula
Now substitute \(2^8\) into the formula:
\[ S_8 = 3 \frac{256 - 1}{1} = 3 \times 255 \]

### Step 6: Calculate the final sum
Now calculate \(3 \times 255\):
\[ S_8 = 765 \]

### Final Answer
Thus, the sum of the first 8 terms of the G.P. is:
\[ \boxed{765} \]