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.

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