If `a_(1),a_(2),a_(3),…` are in `G.P.`, where `a_(i) in C` (where `C` satands for set of complex numbers) having `r` as common ratio such that `sum_(k=1)^(n)a_(2k-1)=sum_(k=1)^(n)a_(2k+3) ne 0` , then the number of possible values of `r` is

Options

A. `2`
B. `3`
C. `4`
D. `5`

Explanation

To solve the problem, we start with the given information about the geometric progression (G.P.) and the sums of specific terms.

### Step 1: Understand the terms in the G.P.
Let the first term of the G.P. be \( a_1 \) and the common ratio be \( r \). The terms can be expressed as:
- \( a_1 = a_1 \)
- \( a_2 = a_1 r \)
- \( a_3 = a_1 r^2 \)
- \( a_4 = a_1 r^3 \)
- \( a_5 = a_1 r^4 \)
- \( a_6 = a_1 r^5 \)
- and so on.

### Step 2: Write the sums
We need to find the sums of the odd-indexed and even-indexed terms:
1. The sum of the first \( n \) odd-indexed terms:
\[ S_{\text{odd}} = a_1 + a_3 + a_5 + \ldots + a_{2n-1} = a_1 + a_1 r^2 + a_1 r^4 + \ldots + a_1 r^{2(n-1)} \]
This is a geometric series with first term \( a_1 \) and common ratio \( r^2 \):
\[ S_{\text{odd}} = a_1 \frac{1 - (r^2)^n}{1 - r^2} = a_1 \frac{1 - r^{2n}}{1 - r^2} \]

2. The sum of the first \( n \) even-indexed terms:
\[ S_{\text{even}} = a_2 + a_4 + a_6 + \ldots + a_{2n} = a_1 r + a_1 r^3 + a_1 r^5 + \ldots + a_1 r^{2n-1} \]
This is also a geometric series with first term \( a_1 r \) and common ratio \( r^2 \):
\[ S_{\text{even}} = a_1 r \frac{1 - (r^2)^n}{1 - r^2} = a_1 r \frac{1 - r^{2n}}{1 - r^2} \]

### Step 3: Set the sums equal
According to the problem, we have:
\[ S_{\text{odd}} = S_{\text{even}} \]
This gives us the equation:
\[ a_1 \frac{1 - r^{2n}}{1 - r^2} = a_1 r \frac{1 - r^{2n}}{1 - r^2} \]

### Step 4: Simplify the equation
Assuming \( a_1 \neq 0 \) and \( 1 - r^{2n} \neq 0 \) (since the sums are non-zero), we can cancel \( a_1 \) and \( \frac{1 - r^{2n}}{1 - r^2} \):
\[ 1 = r \]

### Step 5: Analyze the case when \( r^4 = 1 \)
The equation \( r^4 = 1 \) implies that \( r \) can take the values of the fourth roots of unity:
\[ r = 1, -1, i, -i \]

### Conclusion
Thus, the number of possible values of \( r \) is 4.