Step-by-Step Solution:

1. Define the terms of the arithmetic progressions (APs). Let the first AP be $l_1, l_2, ..., l_100$ with first term $l_1$ and common difference $d_1$. The i-th term is given by $l_i = l_1 + (i-1)d_1$. Let the second AP be $w_1, w_2, ..., w_100$ with first term $w_1$ and common difference $d_2$. The i-th term is given by $w_i = w_1 + (i-1)d_2$.

2. Define the area of the rectangles. For each i, the rectangle $R_i$ has length $l_i$ and width $w_i$. The area $A_i$ is given by $A_i = l_i w_i$.

3. Use the given condition $A_{51} - A_{50} = 1000$. We have $A_{51} = l_{51}w_{51}$ and $A_{50} = l_{50}w_{50}$. We can express $l_{51}$ and $w_{51}$ in terms of $l_{50}$ and $w_{50}$: $l_{51} = l_{50} + d_1$ $w_{51} = w_{50} + d_2$ Now, substitute these into the expression for the difference in areas: $A_{51} - A_{50} = (l_{50} + d_1)(w_{50} + d_2) - l_{50}w_{50}$ $= (l_{50}w_{50} + l_{50}d_2 + w_{50}d_1 + d_1d_2) - l_{50}w_{50}$ $= l_{50}d_2 + w_{50}d_1 + d_1d_2$ We are given that this difference is 1000: $l_{50}d_2 + w_{50}d_1 + d_1d_2 = 1000$

4. Express $l_{50}$ and $w_{50}$ in terms of the first terms. $l_{50} = l_1 + (50-1)d_1 = l_1 + 49d_1$ $w_{50} = w_1 + (50-1)d_2 = w_1 + 49d_2$ Substitute these into the equation from Step 3: $(l_1 + 49d_1)d_2 + (w_1 + 49d_2)d_1 + d_1d_2 = 1000$ $l_1d_2 + 49d_1d_2 + w_1d_1 + 49d_1d_2 + d_1d_2 = 1000$ $l_1d_2 + w_1d_1 + (49 + 49 + 1)d_1d_2 = 1000$ $l_1d_2 + w_1d_1 + 99d_1d_2 = 1000$

5. Use the given relation $d_1d_2 = 10$. Substitute $d_1d_2 = 10$ into the equation from Step 4: $l_1d_2 + w_1d_1 + 99(10) = 1000$ $l_1d_2 + w_1d_1 + 990 = 1000$ $l_1d_2 + w_1d_1 = 10$

6. Calculate the required value $A_{100} - A_{90}$. $A_{100} = l_{100}w_{100} = (l_1 + 99d_1)(w_1 + 99d_2)$ $A_{90} = l_{90}w_{90} = (l_1 + 89d_1)(w_1 + 89d_2)$ $A_{100} - A_{90} = (l_1 + 99d_1)(w_1 + 99d_2) - (l_1 + 89d_1)(w_1 + 89d_2)$ Expand the products: $A_{100} = l_1w_1 + 99l_1d_2 + 99w_1d_1 + 99^2d_1d_2$ $A_{90} = l_1w_1 + 89l_1d_2 + 89w_1d_1 + 89^2d_1d_2$ Subtracting $A_{90}$ from $A_{100}$: $A_{100} - A_{90} = (99-89)l_1d_2 + (99-89)w_1d_1 + (99^2 - 89^2)d_1d_2$ $= 10(l_1d_2 + w_1d_1) + (99-89)(99+89)d_1d_2$ $= 10(l_1d_2 + w_1d_1) + (10)(188)d_1d_2$

7. Substitute the known values to find the final answer. From Step 5, we know $l_1d_2 + w_1d_1 = 10$. We are given $d_1d_2 = 10$. $A_{100} - A_{90} = 10(10) + (10)(188)(10)$ = 100 + 18800 = 18900

Thus, the value of $A_{100} - A_{90}$ is 18900.