If the first term of a G.P is 27 and 8th term is `1/81`, then the sum of its first 10 terms is (i) `27/2(1-1/(3^(10)))` (ii) `81/2(1/(3^(10))-1)` (iii) `27/2(1/(3^(10))-1)` (iv) `81/2(1-1/(3^(10)))`

Options

A. `27/2(1-1/(3^(10)))`
B. `81/2(1/(3^(10))-1)`
C. `27/2(1/(3^(10))-1)`
D. `81/2(1-1/(3^(10)))`

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given values
We are given:
- The first term \( a = 27 \)
- The 8th term \( a_8 = \frac{1}{81} \)

### Step 2: Use the formula for the nth term of a G.P.
The formula for the nth term of a geometric progression (G.P.) is given by:
\[ a_n = a \cdot r^{n-1} \]
For the 8th term, we have:
\[ a_8 = a \cdot r^{8-1} = a \cdot r^7 \]
Substituting the known values:
\[ \frac{1}{81} = 27 \cdot r^7 \]

### Step 3: Solve for the common ratio \( r \)
Rearranging the equation to find \( r^7 \):
\[ r^7 = \frac{1}{81 \cdot 27} \]
Calculating \( 81 \cdot 27 \):
\[ 81 = 3^4 \quad \text{and} \quad 27 = 3^3 \]
Thus:
\[ 81 \cdot 27 = 3^4 \cdot 3^3 = 3^{4+3} = 3^7 \]
Now substituting back:
\[ r^7 = \frac{1}{3^7} = 3^{-7} \]
Taking the 7th root:
\[ r = 3^{-1} = \frac{1}{3} \]

### Step 4: 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 \( S_n \) of a G.P. is:
\[ S_n = a \cdot \frac{1 - r^n}{1 - r} \quad \text{(for } r < 1\text{)} \]
Substituting \( n = 10 \), \( a = 27 \), and \( r = \frac{1}{3} \):
\[ S_{10} = 27 \cdot \frac{1 - \left(\frac{1}{3}\right)^{10}}{1 - \frac{1}{3}} \]

### Step 5: Simplify the expression
Calculating \( 1 - \frac{1}{3} \):
\[ 1 - \frac{1}{3} = \frac{2}{3} \]
Now substituting back into the sum formula:
\[ S_{10} = 27 \cdot \frac{1 - \left(\frac{1}{3}\right)^{10}}{\frac{2}{3}} = 27 \cdot \frac{3}{2} \cdot \left(1 - \frac{1}{3^{10}}\right) \]
\[ S_{10} = \frac{81}{2} \left(1 - \frac{1}{3^{10}}\right) \]

### Step 6: Final expression
The final expression for the sum of the first 10 terms is:
\[ S_{10} = \frac{81}{2} \left(1 - \frac{1}{3^{10}}\right) \]

### Conclusion
The correct option is:
(iv) \( \frac{81}{2} \left(1 - \frac{1}{3^{10}}\right) \)