Given the sequence of numbers `x_1, x_2, x_3,...x_1005.` which satisfy `x_1/(x_1+1) = x_2/(x_2+3) = x_3/(x_3+5) = ...= x_1005/(x_1005 + 2009.)` Also, `x_1+x_2+...x_1005 = 2010.` Nature of the sequence is

Options

A. A.P.
B. G.P.
C. A.G.R
D. H.R.

Answer

To solve the problem step by step, we need to analyze the given conditions and derive the nature of the sequence.

### Step 1: Set Up the Equation
We are given that:
\[ \frac{x_1}{x_1 + 1} = \frac{x_2}{x_2 + 3} = \frac{x_3}{x_3 + 5} = \ldots = \frac{x_{1005}}{x_{1005} + 2009} \]
Let this common value be \( k \). Thus, we can write:
\[ \frac{x_n}{x_n + (2n - 1)} = k \quad \text{for } n = 1, 2, \ldots, 1005 \]

### Step 2: Express Each \( x_n \) in Terms of \( k \)
From the equation:
\[ x_n = k(x_n + (2n - 1)) \]
Rearranging gives:
\[ x_n - kx_n = k(2n - 1) \]
\[ x_n(1 - k) = k(2n - 1) \]
Thus,
\[ x_n = \frac{k(2n - 1)}{1 - k} \]

### Step 3: Sum the Sequence
Now, we need to find the sum:
\[ x_1 + x_2 + \ldots + x_{1005} = 2010 \]
Substituting the expression for \( x_n \):
\[ \sum_{n=1}^{1005} x_n = \sum_{n=1}^{1005} \frac{k(2n - 1)}{1 - k} \]
This can be simplified as:
\[ \frac{k}{1 - k} \sum_{n=1}^{1005} (2n - 1) \]
The sum \( \sum_{n=1}^{1005} (2n - 1) \) is the sum of the first 1005 odd numbers, which is:
\[ 1005^2 \]
Thus,
\[ \sum_{n=1}^{1005} x_n = \frac{k}{1 - k} \cdot 1005^2 \]

### Step 4: Set Up the Equation for the Sum
Setting this equal to 2010 gives:
\[ \frac{k \cdot 1005^2}{1 - k} = 2010 \]
Cross-multiplying yields:
\[ k \cdot 1005^2 = 2010(1 - k) \]
\[ k \cdot 1005^2 + 2010k = 2010 \]
\[ k(1005^2 + 2010) = 2010 \]
Thus,
\[ k = \frac{2010}{1005^2 + 2010} \]

### Step 5: Analyze the Nature of the Sequence
Now, substituting \( k \) back into the expression for \( x_n \):
\[ x_n = \frac{\frac{2010}{1005^2 + 2010}(2n - 1)}{1 - \frac{2010}{1005^2 + 2010}} = \frac{2010(2n - 1)}{(1005^2 + 2010) - 2010} \]
This simplifies to:
\[ x_n = \frac{2010(2n - 1)}{1005^2} \]
This shows that \( x_n \) is a linear function of \( n \), indicating that the sequence is an **Arithmetic Progression (AP)**.

### Conclusion
The nature of the sequence \( x_1, x_2, \ldots, x_{1005} \) is that it forms an **Arithmetic Progression (AP)**.