Question

Let T_r be the r^th term of an AP, where the first term is 'a' and common difference is 'd'. If for some positive integers m ≠ n, T_m = 1/n and T_n = 1/m, then a - d equals:

Answer 1

Correct Answer - Option 1 : 0

Formula used: 

For an AP, with first term 'a' and common difference 'd', n^th term is given by

T_n = a + (n - 1)d

Calculation: 

Given that, first term 'a' and common difference 'd' then

T_m = a + (m - 1)d     

But according to question, T_m = 1/n

⇒ a + (m - 1)d = 1/n        -----(1)

T_n = a + (n - 1)d

But according to question, Tn = 1/m

⇒ a + (n - 1)d = 1/m       -----(2)

Substracting equation (1) and (2), we will get

a + (n - 1)d - [a + (m - 1)d] = 1/m - 1/n

⇒ d(n - m) = \(\frac{n\ -\ m}{mn}\)

⇒ d = 1/mn      

From equation (1) 

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

⇒ a = \(\frac{1}{n}\ -\ \frac{(m \ -\ 1)}{mn}\) 

⇒ a = \(\frac{m\ -\ m\ +\ 1}{mn}\ =\ \frac{1}{mn}\) 

Therefore 

a - d = \(\frac{1}{mn}\ -\ \frac{1}{mn}\ =\ 0\)

Hence, option 1 is correct.