Given the sequence of numbers `x_(1),x_(2),x_(3),x_(4),….,x_(2005)`, `(x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+3)=(x_(3))/(x_(3)+5)=...=(x_(2005))/(x_(2005)+4009)`, the nature of the sequence is

Options

A. `A.P.`
B. `G.P.`
C. `H.P.`
D. None of these

Explanation

To determine the nature of the sequence \( x_1, x_2, x_3, \ldots, x_{2005} \) given the condition

\[ \frac{x_1}{x_1 + 1} = \frac{x_2}{x_2 + 3} = \frac{x_3}{x_3 + 5} = \ldots = \frac{x_{2005}}{x_{2005} + 4009}, \]

we will follow these steps:

### Step 1: Set the common ratio
Let \( k \) be the common value of the fractions:

\[ k = \frac{x_1}{x_1 + 1} = \frac{x_2}{x_2 + 3} = \frac{x_3}{x_3 + 5} = \ldots \]

### Step 2: Express \( x_1 \) in terms of \( k \)
From the first equation, we can express \( x_1 \):

\[ k(x_1 + 1) = x_1 \implies kx_1 + k = x_1 \implies kx_1 - x_1 = -k \implies x_1(k - 1) = -k \implies x_1 = \frac{-k}{k - 1}. \]

### Step 3: Express \( x_2 \) in terms of \( k \)
Now, using the second equation:

\[ k = \frac{x_2}{x_2 + 3} \implies k(x_2 + 3) = x_2 \implies kx_2 + 3k = x_2 \implies kx_2 - x_2 = -3k \implies x_2(k - 1) = -3k \implies x_2 = \frac{-3k}{k - 1}. \]

### Step 4: Express \( x_3 \) in terms of \( k \)
Using the third equation:

\[ k = \frac{x_3}{x_3 + 5} \implies k(x_3 + 5) = x_3 \implies kx_3 + 5k = x_3 \implies kx_3 - x_3 = -5k \implies x_3(k - 1) = -5k \implies x_3 = \frac{-5k}{k - 1}. \]

### Step 5: Identify the pattern
Now we have:

- \( x_1 = \frac{-k}{k - 1} \)
- \( x_2 = \frac{-3k}{k - 1} \)
- \( x_3 = \frac{-5k}{k - 1} \)

We can observe that:

\[ x_2 = 3x_1 \quad \text{and} \quad x_3 = 5x_1. \]

### Step 6: Generalize the expression
Continuing this pattern, we can see that:

\[ x_n = (2n - 1)x_1 \quad \text{for } n = 1, 2, 3, \ldots, 2005. \]

### Step 7: Conclusion
Since the sequence can be expressed in the form \( x_n = (2n - 1)x_1 \), it is evident that the sequence is an arithmetic progression (AP) with a common difference of \( 2x_1 \).

Thus, the nature of the sequence is **Arithmetic Progression (AP)**.

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