The nth term of a G.P. is `3cdot(-2)^(n).` Find the sum of its 7 terms.

Explanation

To find the sum of the first 7 terms of the geometric progression (G.P.) given by the nth term \( a_n = 3 \cdot (-2)^n \), we will follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a_1 \) of the G.P. can be found by substituting \( n = 1 \):
\[ a_1 = 3 \cdot (-2)^1 = 3 \cdot (-2) = -6 \]
The common ratio \( r \) can be found by substituting \( n = 2 \):
\[ a_2 = 3 \cdot (-2)^2 = 3 \cdot 4 = 12 \]
Now, we can find the common ratio:
\[ r = \frac{a_2}{a_1} = \frac{12}{-6} = -2 \]

### Step 2: 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:
\[ S_n = a_1 \cdot \frac{1 - r^n}{1 - r} \quad \text{(for } r \neq 1\text{)} \]
Here, \( n = 7 \), \( a_1 = -6 \), and \( r = -2 \).

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ S_7 = -6 \cdot \frac{1 - (-2)^7}{1 - (-2)} \]

### Step 4: Calculate \( (-2)^7 \)
Calculating \( (-2)^7 \):
\[ (-2)^7 = -128 \]
So,
\[ 1 - (-2)^7 = 1 - (-128) = 1 + 128 = 129 \]

### Step 5: Calculate \( 1 - (-2) \)
Calculating \( 1 - (-2) \):
\[ 1 - (-2) = 1 + 2 = 3 \]

### Step 6: Substitute back into the sum formula
Now substituting back into the sum formula:
\[ S_7 = -6 \cdot \frac{129}{3} \]

### Step 7: Simplify the expression
Calculating \( \frac{129}{3} \):
\[ \frac{129}{3} = 43 \]
Now, substituting this back:
\[ S_7 = -6 \cdot 43 = -258 \]

### Final Answer
Thus, the sum of the first 7 terms of the G.P. is:
\[ \boxed{-258} \]

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