If `(a+(1)/(a))^(2)=3` then what is the value of `1+a^(6)+a^(12) +a^(18) +a^(84) +a^(90) +a^(200) +a^(206)?`

Options

A. 1
B. 0
C. 8
D. `2a^(2)`

Answer

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

\[ \left(a + \frac{1}{a}\right)^2 = 3 \]

### Step 1: Expand the left side using the identity
Using the identity \((x + y)^2 = x^2 + y^2 + 2xy\), we can rewrite the equation as:

\[ a^2 + \frac{1}{a^2} + 2 = 3 \]

### Step 2: Isolate \(a^2 + \frac{1}{a^2}\)
Subtract 2 from both sides:

\[ a^2 + \frac{1}{a^2} = 3 - 2 = 1 \]

### Step 3: Find \(a^4 + \frac{1}{a^4}\)
Now, we can use the identity again to find \(a^4 + \frac{1}{a^4}\):

\[ \left(a^2 + \frac{1}{a^2}\right)^2 = a^4 + \frac{1}{a^4} + 2 \]

Substituting \(a^2 + \frac{1}{a^2} = 1\):

\[ 1^2 = a^4 + \frac{1}{a^4} + 2 \]

This simplifies to:

\[ 1 = a^4 + \frac{1}{a^4} + 2 \]

Subtracting 2 from both sides gives:

\[ a^4 + \frac{1}{a^4} = 1 - 2 = -1 \]

### Step 4: Find \(a^6 + \frac{1}{a^6}\)
Next, we can find \(a^6 + \frac{1}{a^6}\) using the identity:

\[ a^6 + \frac{1}{a^6} = (a^4 + \frac{1}{a^4})(a^2 + \frac{1}{a^2}) - (a^2 + \frac{1}{a^2}) \]

Substituting the values we have:

\[ a^6 + \frac{1}{a^6} = (-1)(1) - 1 = -1 - 1 = -2 \]

### Step 5: Calculate \(1 + a^6 + a^{12} + a^{18} + a^{84} + a^{90} + a^{200} + a^{206}\)
Now, we need to find the value of:

\[ 1 + a^6 + a^{12} + a^{18} + a^{84} + a^{90} + a^{200} + a^{206} \]

We know \(a^6 + \frac{1}{a^6} = -2\), thus:

\[ a^{12} = (a^6)^2 \quad \text{and} \quad a^{18} = a^{12} \cdot a^6 \]

Using \(a^6 = -2\):

\[ a^{12} = (-2)^2 = 4 \]
\[ a^{18} = a^{12} \cdot a^6 = 4 \cdot (-2) = -8 \]

Now, we can find \(a^{84}, a^{90}, a^{200}, a^{206}\) by recognizing that they can be expressed in terms of \(a^6\):

- \(a^{84} = (a^6)^{14} = (-2)^{14} = 16384\)
- \(a^{90} = (a^6)^{15} = (-2)^{15} = -32768\)
- \(a^{200} = (a^6)^{33} = (-2)^{33} = -8589934592\)
- \(a^{206} = (a^6)^{34} = (-2)^{34} = 17179869184\)

### Final Calculation
Now we can sum everything:

\[ 1 + (-2) + 4 + (-8) + 16384 + (-32768) + (-8589934592) + 17179869184 \]

Calculating this step-by-step:

1. \(1 - 2 = -1\)
2. \(-1 + 4 = 3\)
3. \(3 - 8 = -5\)
4. \(-5 + 16384 = 16379\)
5. \(16379 - 32768 = -16389\)
6. \(-16389 - 8589934592 = -8589934581\)
7. \(-8589934581 + 17179869184 = 8589934583\)

Thus, the final answer is:

\[ \boxed{8589934583} \]