If an A.P. consists of n terms with first term a and nth term l

Solution:

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

Here,

The first term (a) = a

The last term (an) = l

Now, as we know,

$a_{n}=a+(n-1) d$

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

$a_{m}=a+(m-1) d$

$=a+m d-d$........(1)

Similarly, for the mth term from the end, we can take l as the first term.

So, we get,

$a_{m}=l-(m-1) d$

$=l-m d+d$.........(2)

Now, we need to prove $a_{n}+a_{m^{\prime}}=a+l$

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

$a_{n}+a_{m^{\prime}}=(a+m d-d)+(l-m d+d)$

$=a+m d-d+l-m d+d$

 

$=a+l$

Therefore, $a_{m}+a_{w^{\prime}}=a+l$