Which term of the sequence $( - 8 + 18i),\,( - 6 + 15i),$ $( - 4 + 12i)$ $,......$ is purely imaginary

For $\mathrm{x} \geq 0$, the least value of $\mathrm{K}$, for which $4^{1+\mathrm{x}}+4^{1-\mathrm{x}}$, $\frac{\mathrm{K}}{2}, 16^{\mathrm{x}}+16^{-\mathrm{x}}$ are three consecutive terms of an $A.P.$ is equal to :

The sum of numbers from $250$ to $1000$ which are divisible by $3$ is

If $\tan \,n\theta = \tan m\theta $, then the different values of $\theta $ will be in

The sums of $n$ terms of two arithmetic progressions are in the ratio $5 n+4: 9 n+6 .$ Find the ratio of their $18^{\text {th }}$ terms.

Different $A.P.$'s are constructed with the first term $100$,the last term $199$,And integral common differences. The sum of the common differences of all such, $A.P$'s having at least $3$ terms and at most $33$ terms is.