If `x=(-1)^(1)+(-1)^(2)+…+(-1)^(2009)` and `Y=(-1)^(1)-(-1)^(2)+(-1)^(3)-(-1)^(4)+……+(-1)^(2009)`, then find x-y

Options

A. 2009
B. 2008
C. 0
D. 1004

Answer

To solve the problem, we need to calculate the values of \( x \) and \( y \) and then find \( x - y \).

### Step 1: Calculate \( x \)

Given:
\[ x = (-1)^{1} + (-1)^{2} + (-1)^{3} + \ldots + (-1)^{2009} \]

- The powers of \(-1\) alternate between \(-1\) (for odd powers) and \(1\) (for even powers).
- The odd powers from \(1\) to \(2009\) are \(1, 3, 5, \ldots, 2009\) (total of 1005 terms).
- The even powers from \(2\) to \(2008\) are \(2, 4, 6, \ldots, 2008\) (total of 1004 terms).

Thus, we can express \( x \) as:
\[ x = (1005 \text{ terms of } -1) + (1004 \text{ terms of } 1) \]
\[ x = -1005 + 1004 = -1 \]

### Step 2: Calculate \( y \)

Given:
\[ y = (-1)^{1} - (-1)^{2} + (-1)^{3} - (-1)^{4} + \ldots + (-1)^{2009} \]

- The odd powers contribute \(-1\) and the even powers contribute \(-1\) (since they are subtracted).
- The odd powers from \(1\) to \(2009\) (1005 terms) contribute \(-1\) each.
- The even powers from \(2\) to \(2008\) (1004 terms) contribute \(+1\) each.

Thus, we can express \( y \) as:
\[ y = (1005 \text{ terms of } -1) - (1004 \text{ terms of } 1) \]
\[ y = -1005 - 1004 = -2009 \]

### Step 3: Calculate \( x - y \)

Now we find \( x - y \):
\[ x - y = -1 - (-2009) \]
\[ x - y = -1 + 2009 = 2008 \]

### Final Answer:
\[ x - y = 2008 \]

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