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 the two arithmetic progressions be written as

$$l_i = l_1 + (i-1)d_1 \qquad (i = 1,2,\ldots,100)$$
$$w_i = w_1 + (i-1)d_2 \qquad (i = 1,2,\ldots,100)$$

The area of the $$i^{\text{th}}$$ rectangle is

$$A_i = l_i\,w_i =\bigl(l_1 + (i-1)d_1\bigr)\bigl(w_1 + (i-1)d_2\bigr).$$

Write the difference between successive areas:

$$A_{i+1}-A_i =\bigl(l_i+d_1\bigr)\bigl(w_i+d_2\bigr)-l_i w_i.$$

Expand the right‐hand side:

$$A_{i+1}-A_i = l_i w_i + d_2 l_i + d_1 w_i + d_1 d_2 - l_i w_i$$
$$\qquad = d_2 l_i + d_1 w_i + d_1 d_2.$$

Substitute $$l_i$$ and $$w_i$$:

$$A_{i+1}-A_i = d_2\bigl(l_1+(i-1)d_1\bigr) + d_1\bigl(w_1+(i-1)d_2\bigr) + d_1 d_2$$
$$\qquad = d_2l_1 + d_1 w_1 + d_1 d_2 + 2(i-1)d_1 d_2.$$

Given $$d_1 d_2 = 10,$$ put $$k=10$$ for brevity:

$$A_{i+1}-A_i = d_2l_1 + d_1 w_1 + k + 2(i-1)k.$$

The data $$A_{51}-A_{50}=1000$$ gives, on taking $$i=50$$ in the above expression,

$$1000 = d_2l_1 + d_1 w_1 + k + 2(49)k$$
$$\Rightarrow 1000 = d_2l_1 + d_1 w_1 + 10 + 98\cdot10$$
$$\Rightarrow d_2l_1 + d_1 w_1 + 10 = 20$$
$$\Rightarrow d_2l_1 + d_1 w_1 = 10.$$

Hence the constant term reduces to

$$d_2l_1 + d_1 w_1 + k = 10 + 10 = 20.$$ Therefore

$$A_{i+1}-A_i = 2(i-1)k + 20 = 20(i-1)+20 = 20i.$$

Thus for every $$i\ge 1$$,

$$A_{i+1}-A_i = 20\,i.$$

Now compute $$A_{100}-A_{90}$$ by summing the ten successive differences:

$$A_{100}-A_{90} = \sum_{i=90}^{99} (A_{i+1}-A_i) = 20\sum_{i=90}^{99} i.$$

The required sum of integers is

$$\sum_{i=90}^{99} i = \frac{99\cdot100}{2}-\frac{89\cdot90}{2} = 4950-4005 = 945.$$

Hence

$$A_{100}-A_{90} = 20 \times 945 = 18900.$$

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