Find the sum of G.P. :
`1+3+9+27+ . . . . .. . ` to 12 terms.

Answer

To find the sum of the geometric progression (G.P.) given by the series \(1 + 3 + 9 + 27 + \ldots\) up to 12 terms, we can follow these steps:

### Step 1: Identify the first term (a), common ratio (r), and number of terms (n)
- The first term \(a\) is \(1\).
- The common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{3}{1} = 3\).
- The number of terms \(n\) is given as \(12\).

### 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 geometric progression is given by:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]

### Step 3: Substitute the values into the formula
Substituting \(a = 1\), \(r = 3\), and \(n = 12\) into the formula:
\[ S_{12} = \frac{1(3^{12} - 1)}{3 - 1} \]

### Step 4: Calculate \(3^{12}\)
Now we need to calculate \(3^{12}\):
\[ 3^{12} = 531441 \]

### Step 5: Substitute \(3^{12}\) back into the formula
Now, substituting \(3^{12}\) back into the equation:
\[ S_{12} = \frac{1(531441 - 1)}{2} \]
\[ S_{12} = \frac{531440}{2} \]

### Step 6: Perform the division
Now, perform the division:
\[ S_{12} = 265720 \]

### Final Answer
The sum of the geometric progression up to 12 terms is \(265720\).

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