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`) = 1

Now as we know

`a_n =  a + (n - 1)d`

So for the mth 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 as the first term

SO we get

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

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

No we need to prove `a_m + a_m' = a + l`

So adding (1) and (2) we get

`a_m + 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