The nth term of a G.P. is `3cdot2^(n).` Find the sum of 8 terms of the G.P.

Explanation

To find the sum of the first 8 terms of the geometric progression (G.P.) given by the nth term \( a_n = 3 \cdot 2^n \), we can follow these steps:

### Step 1: Identify the first term (a)
The first term \( a \) of the G.P. can be found by substituting \( n = 1 \) into the formula for the nth term.

\[ a = 3 \cdot 2^1 = 3 \cdot 2 = 6 \]

### Step 2: Identify the common ratio (r)
The common ratio \( r \) can be determined by taking the ratio of the second term to the first term.

To find the second term, substitute \( n = 2 \):

\[ a_2 = 3 \cdot 2^2 = 3 \cdot 4 = 12 \]

Now, calculate the common ratio:

\[ r = \frac{a_2}{a_1} = \frac{12}{6} = 2 \]

### Step 3: Use the formula for the sum of the first n terms of a G.P.
The formula for the sum \( S_n \) of the first \( n \) terms of a G.P. is given by:

\[ S_n = a \cdot \frac{r^n - 1}{r - 1} \]

### Step 4: Substitute the values into the formula
We need to find the sum of the first 8 terms, so we will substitute \( a = 6 \), \( r = 2 \), and \( n = 8 \) into the formula:

\[ S_8 = 6 \cdot \frac{2^8 - 1}{2 - 1} \]

### Step 5: Calculate \( 2^8 \)
Calculate \( 2^8 \):

\[ 2^8 = 256 \]

### Step 6: Substitute back into the sum formula
Now substitute \( 256 \) back into the sum formula:

\[ S_8 = 6 \cdot \frac{256 - 1}{1} = 6 \cdot 255 \]

### Step 7: Calculate the final sum
Now calculate \( 6 \cdot 255 \):

\[ S_8 = 1530 \]

### Final Answer
The sum of the first 8 terms of the G.P. is \( 1530 \).

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