Question

For an A.P., show that (m + n)^th term + (m – n)^th term = 2 × m^thterm

Answer 1

Let a and d be the first term and common difference, respectively.

`\implies` (m + n)^th term = a + (m + n – 1)d  ...(i) 

And (m – n)^th term = a + (m – n – 1)d  ...(ii)

From (i) + (ii), we get

(m + n)^th term + (m – n)^th term

= a + (m + n – 1)d + a + (m – n – 1)d

= a + md + nd – d + a + md – nd – d

= 2a + 2md – 2d

= 2a + (m – 1)2d

= 2[a + (m – 1)d]

= 2 × m^th term

Hence proved.