Show that the sum of ( m+n)th and (m + n)th term of an A.P. is equal to twice the mth term

Show that the sum of ( m+n)th and (m + n)th term of an A.P. is equal t

1.	Show that the sum of ( m+n)th and (m + n)th term of an A.P. is equal to twice the mth term
Answer» Let a and d be the first term and the common difference of the A.P. respectively.It is known that the kth term of an A. P. is given byak = a + (k –1) d∴ am + n = a + (m + n –1) dam – n = a + (m – n –1) dam = a + (m –1) d∴ am + n + am – n = a + (m + n –1) d + a + (m – n –1) d= 2a + (m + n –1 + m – n –1) d= 2a + (2m – 2) d= 2a + 2 (m – 1) d=2 [a + (m – 1) d]= 2amThus, the sum of (m + n)th and (m – n)th terms of an A.P. is equal to twice the mth term.