If $x,y,z$ are in $A.P. $ and ${\tan ^{ - 1}}x,{\tan ^{ - 1}}y$ and ${\tan ^{ - 1}}z$ are also in $A.P.$, then

Write the first five terms of the sequences whose $n^{t h}$ term is $a_{n}=n \frac{n^{2}+5}{4}$

Let $3,7,11,15, \ldots, 403$ and $2,5,8,11, \ldots, 404$ be two arithmetic progressions. Then the sum, of the common terms in them, is equal to.....................

If ${a_1},\;{a_2},\;{a_3}.......{a_n}$ are in $A.P.$, where ${a_i} > 0$ for all $i$, then the value of $\frac{1}{{\sqrt {{a_1}} + \sqrt {{a_2}} }} + \frac{1}{{\sqrt {{a_2}} + \sqrt {{a_3}} }} + $ $........ + \frac{1}{{\sqrt {{a_{n - 1}}}  + \sqrt {{a_n}} }} = $

If the sum of first $n$ terms of an $A.P.$ be equal to the sum of its first $m$ terms, $(m \ne n)$, then the sum of its first $(m + n)$ terms will be

Let $a_1, a_2 , a_3,.....$ be an $A.P$, such that $\frac{{{a_1} + {a_2} + .... + {a_p}}}{{{a_1} + {a_2} + {a_3} + ..... + {a_q}}} = \frac{{{p^3}}}{{{q^3}}};p \ne q$. Then $\frac{{{a_6}}}{{{a_{21}}}}$ is equal to