If `x +1 //x =2,` then the value of `x ^(7) + 2 //x ^(6) ` is equal to ?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem, we start with the given equation:

1. **Given Equation**:
\[ x + \frac{1}{x} = 2 \]

2. **Finding the value of \(x\)**:
To find \(x\), we can rearrange the equation:
\[ x + \frac{1}{x} - 2 = 0 \]
Multiplying through by \(x\) to eliminate the fraction gives:
\[ x^2 - 2x + 1 = 0 \]
This can be factored as:
\[ (x - 1)^2 = 0 \]
Thus, we find:
\[ x - 1 = 0 \implies x = 1 \]

3. **Calculating \(x^7 + \frac{2}{x^6}\)**:
Now we need to find the value of \(x^7 + \frac{2}{x^6}\):
Substituting \(x = 1\):
\[ x^7 = 1^7 = 1 \]
\[ \frac{2}{x^6} = \frac{2}{1^6} = 2 \]
Therefore:
\[ x^7 + \frac{2}{x^6} = 1 + 2 = 3 \]

4. **Final Answer**:
The value of \(x^7 + \frac{2}{x^6}\) is:
\[ \boxed{3} \]