Question

If mth term of an AP is `1/n` and nth term is `1/m` then find the sum of its first mn terms.

Answer 1

Given:

Let the AP have first term a and common difference d.

`T_m = a + (m - 1)d = 1/n` and `T_n = a + (n - 1)d = 1/m`. (Assume m ≠ n.)

Step-wise calculation:

1. Subtract the two equations:

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

⇒ `d = 1/(mn)`

2. Substitute d into T_m or T_n to find a: 

`a + (m - 1)/(mn) = 1/n` 

⇒ `a = 1/n - (m - 1)/(mn)`

⇒ `a= 1/(mn)`

Thus, `a = d = 1/(mn)`.

3. Sum of first mn terms:

`S_(mn) = (mn)/2 [2a + (mn - 1)d]` 

= `(mn)/2 [2/(mn) + (mn - 1)/(mn)]` 

= `(mn)/2 xx (mn + 1)/(mn)` 

= `(mn + 1)/2`

The sum of the first mn terms is `(mn + 1)/2`.