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 ....... .

If ${S_1},\;{S_2},\;{S_3},...........{S_m}$ are the sums of $n$ terms of $m$ $A.P.'s$ whose first terms are $1,\;2,\;3,\;...............,m$ and common differences are $1,\;3,\;5,\;...........2m - 1$ respectively, then ${S_1} + {S_2} + {S_3} + .......{S_m} = $

The sum of all the elements of the set $\{\alpha \in\{1,2, \ldots, 100\}: \operatorname{HCF}(\alpha, 24)=1\}$ is

If $a,b,c,d,e$ are in $A.P.$ then the value of $a + b + 4c$ $ - 4d + e$ in terms of $a$, if possible is

If ${a_1},\,{a_2},....,{a_{n + 1}}$ are in $A.P.$, then $\frac{1}{{{a_1}{a_2}}} + \frac{1}{{{a_2}{a_3}}} + ..... + \frac{1}{{{a_n}{a_{n + 1}}}}$ is

If $\frac{1}{{p + q}},\;\frac{1}{{r + p}},\;\frac{1}{{q + r}}$ are in $A.P.$, then