Question

If p times pth term of an A.P is equal to q times of qth term, find (p+q)th term of the ap

Answer 1

Given pth term = 1/q
That is ap = a + (p - 1)d = 1/q
aq + (pq - q)d = 1  --- (1)
Similarly, we get ap + (pq - p)d = 1  --- (2)
From (1) and (2), we get
aq + (pq - q)d = ap + (pq - p)d 
aq - ap = d[pq - p - pq + q]
a(q - p) = d(q - p)
Therefore, a = d
Equation (1) becomes,
dq + pqd - dq = 1  
d = 1/pq
Hence a = 1/pq
Consider, Spq = (pq/2)[2a + (pq - 1)d]
                    = (pq/2)[2(1/pq) + (pq - 1)(1/pq)]
                    = (1/2)[2 + pq - 1]
                    = (1/2)[pq + 1]