Question

Mark the correct alternative in the following question:

If in an A.P., the pth term is q and (p + q)th term is zero, then the qth term is

(a) $-$p (b) p (c) p + q (d) p $-$ q

Answer 1

$$\begin{gathered} \mathrm{As},a_p=q \\ \Rightarrow a+\left(p-1\right)d=q.....\left(\mathrm{i}\right) \\ \mathrm{Also},a_{\left(p+q\right)}=0 \\ \Rightarrow a+\left(p+q-1\right)d=0.....\left(\mathrm{ii}\right) \\ \mathrm{Subtracting}\left(\mathrm{i}\right)\mathrm{from}\left(\mathrm{ii}\right),\mathrm{we}\mathrm{get} \\ a+\left(p+q-1\right)d-a-\left(p-1\right)d=0-q \\ \Rightarrow \left(p+q-1-p+1\right)d=-q \\ \Rightarrow qd=-q \\ \Rightarrow d=\frac{-q}{q} \\ \Rightarrow d=-1 \\ \mathrm{Substituting}d=-1\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get} \\ a+\left(p-1\right)\times \left(-1\right)=q \\ \Rightarrow a-p+1=q \\ \Rightarrow a=p+q-1 \\ \mathrm{Now}, \\ {a}_q=a+\left(q-1\right)d \\ =p+q-1+\left(q-1\right)\times \left(-1\right) \\ =p+q-1-q+1 \\ =p \end{gathered}$$

Hence, the correct alternative is option (b).