Consider a sequence whose sum of first $n$ -terms is given by $S_n = 4n^2 + 6n, n \in N$, then $T_{15}$ of this sequence is -

Let $S_{n}$ denote the sum of first $n$-terms of an arithmetic progression. If $S_{10}=530, S_{5}=140$, then $\mathrm{S}_{20}-\mathrm{S}_{6}$ is equal to :

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 ....... .

The number of terms common between the two series $2 + 5 + 8 +.....$ upto $50$ terms and the series $3 + 5 + 7 + 9.....$ upto $60$ terms, is

Let ${a_1},{a_2},.......,{a_{30}}$ be an $A.P.$, $S = \sum\limits_{i = 1}^{30} {{a_i}} $ and $T = \sum\limits_{i = 1}^{15} {{a_{2i - 1}}} $.If ${a_5} = 27$ and $S - 2T = 75$ , then $a_{10}$ is equal to

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  . . .