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. . . . . 

The sum of the common terms of the following three arithmetic progressions.

$3,7,11,15,……………….,399$

$2,5,8,11,…………,359$ and

$2,7,12,17,………..,197$, is equal to $…………….$.

If $\frac{1}{{b – c}},\;\frac{1}{{c – a}},\;\frac{1}{{a – b}}$ be consecutive terms of an $A.P.$, then ${(b – c)^2},\;{(c – a)^2},\;{(a – b)^2}$ will be in

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=\frac{n}{n+1}$

If $\frac{{{a^{n + 1}} + {b^{n + 1}}}}{{{a^n} + {b^n}}}$ be the $A.M.$ of $a$ and $b$, then $n=$