Question:
If an A.P. consists of n terms with first term a and nth term l show that the sum of the mth term from the beginning and the mth term from the end is (a + 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, $a_{m}+a_{w^{\prime}}=a+l$
Hence proved