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

Let $a_n, n \geq 1$, be an arithmetic progression with first term $2$ and common difference $4$ . Let $M_n$ be the average of the first $n$ terms. Then the sum $\sum \limits_{n=1}^{10} M_n$ is

Find the $25^{th}$ common term of the following $A.P.'s$

$S_1 = 1, 6, 11, .....$

$S_2 = 3, 7, 11, .....$

If ${a_1},\;{a_2},\,{a_3},......{a_{24}}$ are in arithmetic progression and ${a_1} + {a_5} + {a_{10}} + {a_{15}} + {a_{20}} + {a_{24}} = 225$, then ${a_1} + {a_2} + {a_3} + ........ + {a_{23}} + {a_{24}} = $

If $a_1 , a_2, a_3, . . . . , a_n, ....$ are in $A.P.$ such that $a_4 - a_7 + a_{10}\, = m$, then the sum of first $13$ terms of this $A.P.$, is .............. $\mathrm{m}$

Let $a , b , c$ be in arithmetic progression. Let the centroid of the triangle with vertices $( a , c ),(2, b)$ and $(a, b)$ be $\left(\frac{10}{3}, \frac{7}{3}\right)$. If $\alpha, \beta$ are the roots of the equation $ax ^{2}+ bx +1=0$, then the value of $\alpha^{2}+\beta^{2}-\alpha \beta$ is ....... .