In a geometrical progression first term and common ratio are both `(1)/(3) (1 sqrt"" 3 + i)` . Then the absolute value of the nth term of the progression is

Options

A. `2^(n)`
B. `4^(n)`
C. 1
D. none of these

Answer

To find the absolute value of the nth term of the given geometric progression, we can follow these steps:

### Step 1: Identify the first term and common ratio
The first term \( A \) and the common ratio \( r \) are both given as:
\[ A = \frac{1}{3} (\sqrt{3} + i) \]
\[ r = \frac{1}{3} (\sqrt{3} + i) \]

### Step 2: Write the formula for the nth term of a geometric progression
The nth term \( T_n \) of a geometric progression can be calculated using the formula:
\[ T_n = A \cdot r^{n-1} \]

### Step 3: Substitute the values of \( A \) and \( r \)
Substituting the values of \( A \) and \( r \) into the formula:
\[ T_n = \left(\frac{1}{3} (\sqrt{3} + i)\right) \cdot \left(\frac{1}{3} (\sqrt{3} + i)\right)^{n-1} \]

### Step 4: Simplify the expression
We can rewrite the nth term as:
\[ T_n = \frac{1}{3} (\sqrt{3} + i) \cdot \left(\frac{1}{3} (\sqrt{3} + i)\right)^{n-1} = \frac{(\sqrt{3} + i)^n}{3^n} \]

### Step 5: Find the absolute value of \( T_n \)
To find the absolute value (modulus) of \( T_n \):
\[ |T_n| = \left| \frac{(\sqrt{3} + i)^n}{3^n} \right| = \frac{|(\sqrt{3} + i)^n|}{|3^n|} \]

### Step 6: Calculate the modulus of \( \sqrt{3} + i \)
The modulus of a complex number \( a + bi \) is given by:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
For \( \sqrt{3} + i \):
\[ |\sqrt{3} + i| = \sqrt{(\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]

### Step 7: Substitute back into the modulus equation
Now substituting back:
\[ |T_n| = \frac{|(\sqrt{3} + i)|^n}{3^n} = \frac{2^n}{3^n} = \left(\frac{2}{3}\right)^n \]

### Conclusion
Thus, the absolute value of the nth term of the progression is:
\[ |T_n| = \left(\frac{2}{3}\right)^n \]