Let $l_1,l_2,\dots ,l_{100}$ be consecutive terms of an arithmetic progression with common difference $d_1$ and let $w_1,w_2,\dots ,w_{100}$ be consecutive terms of another arithmetic progression with common difference $d_2$, where $d_1{d}_2=10$. For each $i=1,2,\dots ,100$, let $R_i$ be a rectangle with length $l_i$ and width $w_i$ and area $A_i$. If $A_{51}-A_{50}=1000$, then the value of $A_{100}-A_{90}$ is ____________.

The correct answer is 18900.

Step 1: Define the terms of the arithmetic progressions
Let $l_1$ be the first term of the first arithmetic progression with common difference $d_1$.
The $i$-th term of this progression is given by:
$l_i=l_1+(i-1)d_1$

Similarly, let $w_1$ be the first term of the second arithmetic progression with common difference $d_2$.
The $i$-th term of this progression is given by:
$w_i=w_1+(i-1)d_2$

Step 2: Express the area $A_i$ of the rectangle $R_i$
The area $A_i$ is the product of its length $l_i$ and width $w_i$:
$A_i=l_i{w}_i=[l_1+(i-1\left)d_1\right][w_1+(i-1\left)d_2\right]$

Expanding the expression for $A_i$:
$A_i=l_1{w}_1+(i-1)(l_1{d}_2+w_1{d}_1)+{(i-1)}^2{d}_1{d}_2$

Given that $d_1{d}_2=10$, let $K=l_1{d}_2+w_1{d}_1$. Then:
$A_i=l_1{w}_1+(i-1)K+10{(i-1)}^2$

Step 3: Analyze the given condition $A_{51}-A_{50}=1000$
Let us evaluate $A_{n+1}-A_n$ in general:
$A_{n+1}-A_n=[l_1{w}_1+nK+10n^2]-[l_1{w}_1+(n-1)K+10{(n-1)}^2]$
$A_{n+1}-A_n=K+10[n^2-{(n-1)}^2]=K+10(2n-1)$

For $n=50$:
$A_{51}-A_{50}=K+10(2\times 50-1)=K+10\left(99\right)=K+990$

We are given that $A_{51}-A_{50}=1000$:
$K+990=1000\Rightarrow K=10$

Step 4: Calculate the required value $A_{100}-A_{90}$
We can express $A_{100}-A_{90}$ by substituting $i=100$ and $i=90$ into our formula for $A_i$:
$A_{100}-A_{90}=[l_1{w}_1+99K+10{\left(99\right)}^2]-[l_1{w}_1+89K+10{\left(89\right)}^2]$
$A_{100}-A_{90}=(99-89)K+10[{99}^2-{89}^2]$

Using the difference of squares formula $a^2-b^2=(a-b)(a+b)$:
${99}^2-{89}^2=(99-89)(99+89)=10\times 188=1880$

Substituting $K=10$ and the value of ${99}^2-{89}^2$ back into the expression:
$A_{100}-A_{90}=10\left(10\right)+10\left(1880\right)$
$A_{100}-A_{90}=100+18800=18900$

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