Find the area in the 1* quadrant bounded by `[x]+[y]=n`, where `n in N and y=k`(where `k in n AA k<=n+1`), where [.] denotes the greatest integer less than or equal to x.

Answer

To solve the problem of finding the area in the first quadrant bounded by the equation \([x] + [y] = n\) and the line \(y = k\) (where \(k \in \mathbb{N}\) and \(k \leq n + 1\)), we can follow these steps:

### Step 1: Understand the equation \([x] + [y] = n\)

The equation \([x] + [y] = n\) describes a series of line segments in the coordinate plane. Here, \([x]\) and \([y]\) are the greatest integer functions, meaning that \([x]\) is the largest integer less than or equal to \(x\) and \([y]\) is the largest integer less than or equal to \(y\).

### Step 2: Analyze the boundaries

The equation \([x] + [y] = n\) can be rewritten as:
- For \([x] = k\), \([y] = n - k\), where \(k\) can take values from \(0\) to \(n\).

This means that for each integer \(k\) from \(0\) to \(n\), there is a corresponding line segment where:
- \(x\) ranges from \(k\) to \(k + 1\)
- \(y\) ranges from \(n - k\) to \(n - k + 1\)

### Step 3: Find the area bounded by the lines

The area in the first quadrant bounded by these lines can be visualized as a series of rectangles. Each rectangle has a width of \(1\) (from \(k\) to \(k + 1\)) and a height of \(1\) (from \(n - k\) to \(n - k + 1\)).

### Step 4: Calculate the area for \(y = k\)

Now, we need to find the area bounded by the line \(y = k\) for \(k \leq n + 1\).

1. For \(k = 0\), the area is the entire region below the line segment from \((0, n)\) to \((n, 0)\).
2. For \(1 \leq k \leq n\), the area will be bounded by the line \(y = k\) and the segments of the lines defined by \([x] + [y] = n\).

### Step 5: Area Calculation

The area can be calculated as follows:
- The area of each rectangle formed under the line \(y = k\) is given by the height of the rectangle multiplied by the width.
- The total area can be computed by summing the areas of all rectangles formed by the segments of the lines.

The area \(A\) can be expressed as:
\[ A = \sum_{k=0}^{n} (n - k) \cdot 1 \]

### Step 6: Final Calculation

The final area can be simplified:
\[ A = n(n + 1)/2 \]

### Conclusion

Thus, the area in the first quadrant bounded by \([x] + [y] = n\) and \(y = k\) is given by:
\[ \text{Area} = \frac{n(n + 1)}{2} \]