Question

if the m th term of an AP is 1/n andthe n th term is 1/m,show that the sum of mn terms is 1/2(mn+1).

Answer 1

Here we have to find the value of a and d and put them in equation of sum of mn terms.

Answer 2

am=1/n
a+(m-1)d=1/n
an+mnd-nd=1 __(1)

an=1/m
a+(n-1)d=1/m
am+mnd-md=1 __(2)

From (1) and (2)
an+mnd-nd = am+mnd-nd
→ an-nd = am-md
→ n(a-d) = m(a-d)
→ a(n-m) - d(n-m) = 0
→ a(n-m) = d(n-m)
→ a = d

Now, putting value of a in (1)
an+mnd-nd=1
→ dn+mnd-nd=1
→ d=1/mn and a=1/mn

Sum of mn terms = Smn
= mn/2 [2a +(mn-1)d]
= mn/2 [2/mn + (mn-1)mn ]
= mn/2mn [2 + mn-1]
= 1/2 (mn+1)
Hence, proved.
Thanks :)