Question

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

Answer 1

Given:-

• mth term of an AP is 1/n.
• the nth term is 1/m.

To prove:-

• the sum of mn terms is 1/2(mn+1).

step-by-step solution:-

Let mth term of AP be ‘Am’ and nth term of AP be ‘An’

Therefore,

• Am = a+ (m-1)d=1/n ----------(i)

• An = a+(n-1)d=1/m ------------(ii)

Subtracting equation (ii) from (i)

• d[(m - 1) - (n - 1)] = 1/n - 1/m,

• d(m - n) = (m - n)/mn,

• d = 1/mn --------(iii)

Substituting equation (iii) in (i)

• a + (m - 1)/mn = 1/n,

• a = 1/n[1 - (m - 1)/m],

• a = 1/mn --------(iv)

Now Amn i.e the mnth term of AP = a + (mn - 1)d,

Substitute equation (iii) and (iv) in Amn,

• 1/mn + (mn - 1)/mn

• = 1/mn[1 + (mn - 1)]

• = mn/mn = 1,

then the mn term = 1

Sum of MN term:- :-

Amn = mn/2 ( 2/mn + (mn-1)1/mn)

• = 1 + (mn)/2 - 1/2

• = mn /2 + 1/2

• = 1/2 (mn + 1)