Question

if mth term of A. P. is 1/n and nth term is 1/m then find the sum of its first mn term?

Answer 1

m th term of an A.P = 1/n

$\frac{1}{n} = a \: + (m - 1)d$
N th term of an A.P =1/m

$\frac{1}{m} = a \: + (n - 1)d$

Now,
$\frac{1}{n} - \frac{1}{m} = m - n \times d$
m-n/mn=m-n× d

1/mn=d

mn=1/d

From eq(1)

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

a= m-m+1/mn

a = 1/mn

We know that ,

Sum of first mn terms = mn [2a + (mn - 1)d]

= mn [ 2(1/mn) + (mn-1)1/mn]

= mn [ 2/mn+1-1/mn ]

= mn [ 1/mn+1]

= 1+mn .

$there \: fore \: sum \: of \: mn \: terms = 1 + mn \:$