The ratio of the sums of first $n$ even numbers and $n$ odd numbers will be

If $a$ and $b$ are the roots of $x^{2}-3 x+p=0$ and $c, d$ are roots of $x^{2}-12 x+q=0$ where $a, b, c, d$ form a $G.P.$ Prove that $(q+p):(q-p)=17: 15$

If the sum of $n$ terms of an $A.P$. is $2{n^2} + 5n$, then the ${n^{th}}$ term will be

If $1, \log _{10}\left(4^{x}-2\right)$ and $\log _{10}\left(4^{x}+\frac{18}{5}\right)$ are in
arithmetic progression for a real number $x$ then the value of the determinant $\left|\begin{array}{ccc}2\left(x-\frac{1}{2}\right) & x-1 & x^{2} \\ 1 & 0 & x \\ x & 1 & 0\end{array}\right|$ is equal to ...... .

Let the sum of $n, 2 n, 3 n$ terms of an $A.P.$ be $S_{1}, S_{2}$ and $S_{3},$ respectively, show that $S_{3}=3\left(S_{2}-S_{1}\right)$

If $3^{2 \sin 2 \alpha-1},14$ and $3^{4-2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$, then the sixth term of this $A.P.$ is