Question

if the mthterm of an AP is 1/n and nth term is 1/m,them show that the sum of mn term is 1/2(mn+1).

Answer 1

dont know yaar be lated and so sorry for that

Answer 2

am = 1/n
an = 1/m
Smn =?
am = a + (m-1)d = 1/n
an = a + (n-1)d = 1/m

Subtracting the equations we get
dm -d -(dn -d) =1/n-1/m
dm-d-dn+d = m-n/mn (taking LCM)
dm-dn = m-n/mn
d(m-n)= m-n/mn
Therefore, d = 1/mn

Substituting d in any one equation

a + (m-1)*d = 1/n
a + (m-1)*1/mn = 1/n
a + 1/n -1/mn = 1/n
Therefore, a = 1/mn


Now we need Smn

So

Smn = mn/2(2a + (mn-1)d)
= mn/2 (2*1/mn + (mn-1)*1/mn)
=mn/2(2/mn + 1 -1/mn)
= mn/2 (1/mn +1)
=mn/2(mn+1/mn)
= mn+1/2 or 1/2mn
Hence the proof