`100^alpha-199beta=100(100)+99(101)+98(102)+ . . . + 1(199).` then find the slope of line with point `(alpha,beta)` and origin.

Answer

To solve the equation \(100^\alpha - 199\beta = 100(100) + 99(101) + 98(102) + \ldots + 1(199)\) and find the slope of the line passing through the points \((\alpha, \beta)\) and the origin \((0, 0)\), we will follow these steps:

### Step 1: Simplify the Right-Hand Side
The right-hand side of the equation can be expressed as a summation:
\[ \text{RHS} = \sum_{r=0}^{99} (100 - r)(100 + r) \]
This can be simplified using the identity \((a - b)(a + b) = a^2 - b^2\):
\[ \text{RHS} = \sum_{r=0}^{99} (100^2 - r^2) = \sum_{r=0}^{99} 100^2 - \sum_{r=0}^{99} r^2 \]
Calculating \(100^2\):
\[ 100^2 = 10000 \]
Thus,
\[ \text{RHS} = 10000 \cdot 100 - \sum_{r=0}^{99} r^2 \]

### Step 2: Calculate the Summation of Squares
The formula for the sum of the squares of the first \(n\) natural numbers is:
\[ \sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6} \]
For \(n = 99\):
\[ \sum_{r=0}^{99} r^2 = \frac{99(99+1)(2 \cdot 99 + 1)}{6} = \frac{99 \cdot 100 \cdot 199}{6} = 328350 \]

### Step 3: Substitute Back into the Equation
Now substituting back into the equation:
\[ 100^\alpha - 199\beta = 10000 \cdot 100 - 328350 \]
Calculating \(10000 \cdot 100\):
\[ 10000 \cdot 100 = 1000000 \]
Thus,
\[ 100^\alpha - 199\beta = 1000000 - 328350 = 671650 \]

### Step 4: Set Up the Equation
Now we have:
\[ 100^\alpha - 199\beta = 671650 \]

### Step 5: Solve for \(\alpha\) and \(\beta\)
Assuming \(100^\alpha = 671650 + 199\beta\), we can find values for \(\alpha\) and \(\beta\).

Let’s assume \(\alpha = 3\):
\[ 100^3 = 1000000 \]
Then,
\[ 1000000 - 199\beta = 671650 \]
This gives:
\[ 199\beta = 1000000 - 671650 = 328350 \]
Thus,
\[ \beta = \frac{328350}{199} = 1650 \]

### Step 6: Find the Slope
Now we have \((\alpha, \beta) = (3, 1650)\). The slope \(m\) of the line through the points \((0, 0)\) and \((\alpha, \beta)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1650 - 0}{3 - 0} = \frac{1650}{3} = 550 \]

### Final Answer
The slope of the line is:
\[ \boxed{550} \]