The sum of the product taken two at a time of the numbers `1,2,2^2,2^3,.....2^(n-2),2^(n-1)` is (a) `1/3 . 2^(2n) +2/3` (b) `1/3. 2^(2n) -2^n + 1/3` (c) `1/3. 3^(2n) -1/3` (d) none of these

Options

A. `1/3 . 2^(2n) +2/3`
B. `1/3. 2^(2n) -2^n + 1/3`
C. `1/3. 3^(2n) -1/3`
D. `1/3 . 2^(2n) -2^n + 2/3`

Answer

To solve the problem, we need to find the sum of the products taken two at a time from the series: \(1, 2, 2^2, 2^3, \ldots, 2^{n-2}, 2^{n-1}\).

### Step-by-Step Solution:

1. **Identify the series**: The series consists of the terms \(2^0, 2^1, 2^2, \ldots, 2^{n-1}\). This can be rewritten as:
\[ a_1 = 1, \quad a_2 = 2, \quad a_3 = 2^2, \quad \ldots, \quad a_n = 2^{n-1} \]

2. **Calculate the sum of products taken two at a time**: The sum of the products taken two at a time can be expressed as:
\[ S = a_1 a_2 + a_1 a_3 + a_1 a_4 + \ldots + a_{n-1} a_n \]
This can be simplified to:
\[ S = 1 \cdot 2 + 1 \cdot 2^2 + 1 \cdot 2^3 + \ldots + 1 \cdot 2^{n-1} + 2 \cdot 2^2 + 2 \cdot 2^3 + \ldots + 2^{n-2} \cdot 2^{n-1} \]

3. **Rearranging the terms**: We can group the terms based on the powers of 2:
\[ S = \sum_{i=1}^{n-1} 2^i + \sum_{i=2}^{n-1} 2^{2i-1} \]

4. **Calculate the first sum**: The first sum is a geometric series:
\[ \sum_{i=1}^{n-1} 2^i = 2 + 2^2 + 2^3 + \ldots + 2^{n-1} = 2(1 + 2 + 2^2 + \ldots + 2^{n-2}) = 2 \cdot \frac{2^{n-1} - 1}{2 - 1} = 2^{n} - 2 \]

5. **Calculate the second sum**: The second sum can also be simplified:
\[ \sum_{i=2}^{n-1} 2^{2i-1} = 2^3 + 2^5 + \ldots + 2^{2(n-1)-1} = 2^3(1 + 2^2 + 2^4 + \ldots + 2^{2(n-3)}) = 2^3 \cdot \frac{2^{2(n-3)+2} - 1}{2^2 - 1} = \frac{2^{2n-2} - 8}{3} \]

6. **Combine the sums**: Now combine both sums:
\[ S = (2^n - 2) + \frac{2^{2n-2} - 8}{3} \]
Simplifying this gives:
\[ S = \frac{3(2^n - 2) + 2^{2n-2} - 8}{3} = \frac{2^{2n-2} + 3 \cdot 2^n - 14}{3} \]

7. **Final expression**: The final expression can be simplified further, but we can check against the options given in the problem.

### Conclusion:
After evaluating the sums and simplifying, we find that the answer does not match any of the provided options. Therefore, the answer is:
**(d) none of these.**