Find the sum of the G.P. : `2+6+18+54+ . . . . . . . .. +4374`.

Answer

To find the sum of the geometric progression (G.P.) given by \(2 + 6 + 18 + 54 + \ldots + 4374\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the G.P. is:
\[ a = 2 \]
The common ratio \(r\) can be calculated by dividing the second term by the first term:
\[ r = \frac{6}{2} = 3 \]

### Step 2: Determine the number of terms in the G.P.
The \(n\)th term of a G.P. is given by the formula:
\[ T_n = a \cdot r^{n-1} \]
We know that the last term \(T_n\) is 4374. Thus, we can set up the equation:
\[ 4374 = 2 \cdot 3^{n-1} \]
To find \(n\), we first divide both sides by 2:
\[ 2187 = 3^{n-1} \]
Next, we can express 2187 as a power of 3:
\[ 2187 = 3^7 \]
Thus, we have:
\[ 3^{n-1} = 3^7 \implies n - 1 = 7 \implies n = 8 \]

### Step 3: Use the formula for the sum of the first \(n\) terms of a G.P.
The sum \(S_n\) of the first \(n\) terms of a G.P. is given by the formula:
\[ S_n = \frac{a(r^n - 1)}{r - 1} \]
Substituting the values we found:
\[ S_8 = \frac{2(3^8 - 1)}{3 - 1} \]
Calculating \(3^8\):
\[ 3^8 = 6561 \]
Now substituting this back into the sum formula:
\[ S_8 = \frac{2(6561 - 1)}{2} = 6560 \]

### Final Answer
Thus, the sum of the G.P. is:
\[ \boxed{6560} \]