If $\displaystyle \mathrm{p}^{\text {th }}$ term of an A.P. is q and $\displaystyle \mathrm{q}^{\text {th }}$ term is p, then prove that its $\displaystyle \mathrm{n}^{\text {th }}$ term is $\displaystyle (\mathrm{p}+\mathrm{q}-\mathrm{n})$.

\[\begin{aligned} & \mathbf{a}_{\mathbf{p}}=\mathbf{a}+(\mathbf{p}-\mathbf{1}) \mathbf{d}=\mathbf{q} \_ \text {(i) } \\ & \mathbf{a}_{\mathbf{q}}=\mathbf{a}+(\mathbf{q}-1) \mathbf{d}=\mathbf{p} \_\mathbf{( i i )} \\ & \text { Solving (i) and (ii) } \\ & \mathbf{d}=-\mathbf{1}, \mathbf{a}=\mathbf{q}+\mathbf{p}-1 \\ & \mathbf{a}_{\mathbf{n}}=(\mathbf{q}+\mathbf{p}-1)+(\mathbf{n}-1)(-1)=\mathbf{q}+\mathbf{p}-\mathbf{n} \end{aligned} \]