Question

The mth term of an AP is 1/n and nth term is 1/m then show that sum of mn term is 1/2(mn +1)

Answer 1

Question

The mth term of an AP is 1/n and nth term is 1/m then show that sum of mn term is 1/2(mn +1)

Answer:1/2 (mn + 1)

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)

Answer 2

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