Question

If an A.P. consists of n terms with first term a and n^th term `l` show that the sum of the m^th term from the beginning and the m^th term from the end is (a + `l`).

Answer 1

In the given problem, we have an A.P. which consists of n terms.

Here,

The first term (a) = a

The last term `(a_n) = l`

Now, as we know,

`a_n = a + (n - 1)d`

So, for the m^th term from the beginning, we take (n = m)

`a_m = a + (m - 1)d`

= a + md - d ......(1)

Similarly, for the m^th term from the end, we can take `l` as the first term.

So, we get,

`a_m = l - (m -1)d`

= l - md + d ....(2)

Now, we need to prove `a_n + a_(m') = a + l`

So, adding (1) and (2), we get,

`a_n + a_(m') = (a + md - d) + (l - md + d)`

= a + md - d + l - md + d

= a + `l`

Therefore, `a_m + a_m' = a + l`

Hence proved