The first and the last term of an A.P. are a and l respectively. Show

1.	The first and the last term of an A.P. are a and l respectively. Show that the sum of the nth term from the beginning and the nth term from the end is a + l.
Answer» To prove : Sum of the n^th term from the beginning and the nth term from the end is a + l We know, a_n = a + (n – 1)d Where a is first term or a₁ and d is the common difference, and n is any natural number, and n th term from the end is an'= l – (n – 1)d Now, a_n + a_n' = a + (n – 1)d + l – (n – 1)d ⇒ a_n+ a_n' = a + nd – d + l – nd + d ⇒ a_n + an' = a + l Hence Proved.

The first and the last term of an A.P. are a and l respectively. Show that the sum of the nth term from the beginning and the nth term from the end is a + l.

Answer»

To prove :

Sum of the n^th term from the beginning and the nth term from the end is a + l

We know,

a_n = a + (n – 1)d

Where a is first term or a₁ and d is the common difference, and n is any natural number, and n th term from the end is an'= l – (n – 1)d

Now,

a_n + a_n' = a + (n – 1)d + l – (n – 1)d

⇒ a_n+ a_n' = a + nd – d + l – nd + d

⇒ a_n + an' = a + l

Hence Proved.