Let $l_1, l_2, \ldots, l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$, and let $w_1, w_2, \ldots, w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1 d_2 = 10$. For each $i = 1, 2, \ldots, 100$, let $R_i$ be a rectangle with length $l_i$, width $w_i$ and area $A_i$. If $A_{51} - A_{50} = 1000$, then the value of $A_{100} - A_{90}$ is ______.

Let l₁, l₂, …, l₁₀₀ be consecutive terms of an arithmetic progression with common difference d₁, and let w₁, w₂, …, w₁₀₀ be consecutive terms of another arithmetic progression with common difference d₂, where d₁ d₂ = 10. For each i = 1, 2, …, 100, let Rᵢ be a rectangle with length lᵢ, width wᵢ and area Aᵢ. If A₅₁ - A₅₀ = 1000, then the value of A₁₀₀ - A₉₀ is ______.

$\text{Given: } l_i = l_1 + (i-1)d_1, \quad w_i = w_1 + (i-1)d_2$

$A_i = l_i \cdot w_i$

$d_1 d_2 = 10$

$\text{To find: } A_{100} - A_{90} \text{, given } A_{51} - A_{50} = 1000$

$D_i = A_{i+1} - A_i$

$A_{i+1} = (l_i + d_1)(w_i + d_2)$

$D_i = l_i d_2 + w_i d_1 + d_1 d_2$

$D_{i+1} - D_i = d_1 d_2 + d_2 d_1 = 2 d_1 d_2$

$D_{i+1} - D_i = 2 d_1 d_2 = 20$

$D_{50} = A_{51} - A_{50} = 1000$

$D_i = 1000 + 20(i - 50) = 20i$

$A_{100} - A_{90} = \sum_{i=90}^{99} D_i$

$= \sum_{i=90}^{99} 20i = 20(90 + 91 + \cdots + 99)$

$90 + 91 + \cdots + 99 = \dfrac{10}{2}(90 + 99) = 5 \times 189 = 945$

$A_{100} - A_{90} = 20 \times 945$

$\boxed{A_{100} - A_{90} = 18900}$

$D_{50} = 20 \times 50 = 1000 \ \checkmark$