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

$A$ farmer buys a used tractor for $Rs. 12000$. He pays $Rs. 6000$ cash and agrees to pay the balance in annual instalments of $Rs. 500$ plus $12\%$ interest on the unpaid amount. How much will the tractor cost him?

The sum of the series $\frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \dots$ to $9$ terms is

If the roots of the equation $x^3+3px^2+3qx-8=0$ are in an arithmetic progression,then $2p^3-3pq=$

If $S_n = nP + \frac{1}{2}n(n - 1)Q$,where $S_n$ denotes the sum of the first $n$ terms of an $A.P.$,then the common difference is

If $\frac{S_n}{S_m} = \frac{n^4}{m^4}$ (where $S_k$ is the sum of the first $k$ terms of an $A$.$P$. $a_1, a_2, \dots$),then the value of $\frac{a_{m+1}}{a_{n+1}}$ in terms of $m$ and $n$ is: