Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^{2}+n+2 & n^{2} \\ n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If $$\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96$$, then $$n$$ is equal to _____________.

Let the digits a, b, c be in A. P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Two circles in the first quadrant of radii $$r_{1}$$ and $$r_{2}$$ touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line $$x+y=2$$. Then $$r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}$$ is equal to ___________.

A fair $$n(n > 1)$$ faces die is rolled repeatedly until a number less than $$n$$ appears. If the mean of the number of tosses required is $$\frac{n}{9}$$, then $$n$$ is equal to ____________.