Question

in an arithmetic progression, if pth term is1/q and qth term is 1/p,prove that the sum of first pq terms is 1+pq/2,where p is not equal q

Answer 1

let a is the first term and d is the common difference .
according to question,
pth term =1/q
a+(p-1) d=1/q ================(1)
again
qth term =1/p
a+(q-1) d=1/p=================(2)
solve equation (1) and (2)

a=1/pq and d=1/pq
now
Sn=n/2 {2a+(n-1) d}

Spq=pq/2 {2/pq+(pq-1) 1/pq}
=pq/2 {(1+pq)/pq}
=(1+pq)/2