The $2^{nd}, {31}^{st}$ and last term of an $\mathrm{A}.\mathrm{P}.$ are $7\frac{3}{4}, \frac{1}{2}$ and $-6\frac{1}{2}$, respectively, find the first term and the number of terms.

Sum the series

$\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}} \cdots \cdots \cdots$ to $n$ terms

If the $p^{\mathrm{th}}$, $q^{\mathrm{th}}$, $r^{\mathrm{th}}$ terms of an $\mathrm{A}.\mathrm{P}.$ are $a, b, c$, respectively, show that 

$(q-r)a+(r-p)b+(p-q)c=0$