Question

If the mth term of an ap is 1/n and the nth term is 1/m then find the mnth term

Answer 1

am=a+(m-1)d                            ; an=a+(n-1)d

am=1/n                                       an=1/m

therefore, a+(m-1)d=1/n             therefore,a+(n-1)d=1/m

...........eqn 1                                 ........ eqn 2

eqn(1-2)

1/n-1/m=a+(m-1)d-{a+(n-1)d}

taking lcm on lhs,

m-n/mn=a+md-d-a-nd+d

therefore,m-n/mn=md-nd

=>m-n/mn=d(m-n)

1/mn=d(m-n/(m-n)

therefore ,d=1/mn.............eqn 3

eqn 3 in eqn 1

am=a+(m-1)d

1/n=a+(m-1)/mn

a=1/n - (m-1)/mn

taking mn as lcm,

a=m/mn - (m-n)/mn

a=m-(m-n)/mn

a=m-m+n/mn

a=n/mn

tharefore, a=1/m

now taking amn = a+(mn-1)d

amn= 1/mn + mn/mn-1/mn   {substituting  the value of d=1/mn}

amn  = mn/mn

therefore , amn =1

and that's the end of the answer

Hope it helps u.............