Question

The $p^{th}term{T}_p$ of H.P. is $q(q+p)$ and $q^{th}$ term $T_q$ is $p(p+q)$ when p > 1, q > 1, then

• A. $T_{p+q}=pq$
• B. $T_{pq}$ = p + q
• C. $T_{p+q>T_{pq}}$
• D. $T_{pq}>T_{p+q}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $T_{pq}$ = p + q
B $T_{p+q}=pq$
C $T_{p+q>T_{pq}}$

$T_p=\frac{1}{a+(p-1)}d=q(p+q)$ ...(1)
$T_q=\frac{1}{a+(q-1)}d=p(p-q)$ ...(2)
From (1) and (2)
$\therefore \frac{d}{q(a+p-1)}=\frac{d}{p(a+(q-1))}\Rightarrow ap+pq-p=aq+pq-q.\Rightarrow a(p-q)=p-q\Rightarrow a=1.\therefore d=pq(p+q)$
Now
$Tp+q=\frac{1}{a+(p+q-1)}d=\frac{1}{1+p+q-1}pq(p+q)=pq$
$Tpq=\frac{1}{a+(pq-1)}d=\frac{1}{1+pq-1}pq(p+q)=p+q$