If $2x,\;x + 8,\;3x + 1$ are in $A.P.$, then the value of $x$ will be

If three distinct number $a, b, c$ are in $G.P.$ and the equations $ax^2 + 2bc + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root, then which one of the following statements is correct?

For any three positive real numbers $a,b,c$ ; $9\left( {25{a^2} + {b^2}} \right) + 25\left( {{c^2} - 3ac} \right) = 15b\left( {3a + c} \right)$ then

If $n$ be odd or even, then the sum of $n$ terms of the series $1 - 2 + $ $3 - $$4 + 5 - 6 + ......$ will be

If the ratio of the sum of $n$ terms of two $A.P.'s$ be $(7n + 1):(4n + 27)$, then the ratio of their ${11^{th}}$ terms will be

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