If the $j^{th}$ term and $k^{th}$ term of an A.P. are $k$ and $j$ respectively, then ${\left(k+j\right)}^{th}$term is:? 

Let a be the first term and d be the common difference. General term ($n^{th}$ term) of the AP is 

$T_n=a+(n-1)d$

As given, $T_j=a+(j-1)d=k ......\left(1\right)$

${\mathrm{T}}_{\mathrm{k}}=\mathrm{a}+(\mathrm{k}-1)\mathrm{d}=\mathrm{j} .....\left(2\right)$

Subtracting (2) from (1) we get 

$(j-k)d=k-j\Rightarrow d=-1$

On putting $\mathrm{d}=-1$ in (1), we get

$\begin{array}{l}a+(j-1)(-1)=k\\ \Rightarrow a=k+j-1\end{array}$

Now, $T_{k+j}=a+(k+j-1)d=k+j-1+\left[\right(k+j)-1](-1)=(k+j-1)-(k+j-1)=0$