Question

if mth term of an AP is 1/n and nth term is1/m ,then find the sum of its first mn terms

Answer 1

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Answer 2

Given,

the mth term of AP = 1 / n

The nth term of AP = 1 / m

To Find,

The sum of its first mn terms =?

Solution,

From the formula of nth term in AP,

$a_{n} = a +(n-1)d$

For mth term,

$a_{m} = a +(m-1)d = 1 / n\\an + mnd - nd= 1$⇒Equation 1

For nth term.

$a_{n} = a +(n-1)d = 1 / m\\am + mnd -md =1$⇒Equation 2

Equating equations 1 and 2,

am + mnd - md = an + mnd - nd

am - md = an - nd

am - an = md - nd

a(m-n) = d(m-n)

a = d ⇒Equation 3

Putting a =d in equation 1,

an + mnd -nd = 1

dn + mnd - nd = 1

mnd = 1

d = 1 / mn

Also, a  = d = 1/mn

Sum upto mn term,

$s_{mn } = mn(2a + (mn-1)d)/2\\s_{mn } = mn(2/mn+ (mn-1)/mn)/2\\s_{mn } = mn(2+ mn-1)/2mn\\s_{mn } = (1+ mn)/2\\s_{mn } =(1+ mn)/2$

Hence, the sum of its first mn terms is (mn + 1) / 2.