If the pth term of an AP is 1/q and qth term is 1/p find the sun of its first 20 terms

The nth term of an AP = a + (n - 1) d

The pth term of the AP = a + (p - 1) d = 1/q

The qth term of the AP = a + (q - 1) d = 1/p

Sum of nth term = (n/2) (2a + (n - 1) d)

Sum for 20th term = (20/2) (2a + (20 - 1) d) = 10 (2a + 19d)

a + (p - 1) d = 1/q ... (a)

a + (q - 1) d = 1/p ... (b)

Equation (a) - (b)

a - a + (p - 1) d - (q - 1) d = 1/q - 1/p

pd - d - qd + d = (p - q) / pq

pd - qd = (p - q) / pq

(p - q) d = (p - q) / pq

Cancel out (p - q) from both sides

d = 1 / (pq)

Recall a + (p - 1) d = 1/q ... (a)

a + (p - 1) (1/pq) = 1/q

a + (p - 1) / (pq) = 1/q

a = (1/q) - (p - 1) / (pq)

a = (p - (p - 1)) / (pq)

a = (p - p + 1) / (pq)

a = 1 / (pq)

a = 1 / (pq), d = 1 / (pq)

Recall sum of 20 terms:

10 (2a + 19d)

10 (2 * (1 / (pq)) + 19 * (1 / (pq)))

10 (2 / (pq) + 19 / (pq)) = 10 * (2 + 19) / (pq)

= 10*21 / (pq)

= 210 / (pq)

Sum of the 20 terms = 210 / (pq)

Hope this explains it.