Let a be the first term & d be the common difference of the A.P., then
ap = a + (p - 1) d =

1

q

... (i)
aq = a + (q - 1) d =

1

p

... (ii)
From (i) & (ii), we get
(p - 1) d - (q - 1) d =

1

q

−

1

p

⇒ (p - 1 - q + 1) d =

p−q

pq

⇒ (p - q) d =

p−q

pq

⇒ d =

1

pq

... (iii)
Then a +

(p−1)

pq

=

1

q

[From (i) & (iii)]
⇒ a =

1

q

−

p−1

pq

=

p−p+1

pq

=

1

pq

Hence, the sum of first pq terms
Spq =

pq

2

[

2

pq

+

(pq−1)

pq

] =

pq

2

[

(2+pq−1)

pq

] =

pq+1

2

, where p ≠ q.