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Insertion of AP's between 2 Numbers

For an ap if ...

Question

For an ap if tm =1/n and tn=1/m then a-d=

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Solution

let a be the first term and d be the common difference of an AP series

tm = a+ (m-1)d =1/n ….(i) (given)

tn = a+(n-1)d=1/m …...(ii) (given)

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)

Hence a-d = 1/mn - 1/mn = 0 Ans.

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