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 ends is $(a+l)$

Answer 1

Given,

number of terms in A.P is n.

first term of A.P is $a_1=a$

nth term of A.P is $a_n=l$

let d be the common difference

mth term of an A.P from begning is given by

$a_m=a+(m-1)d...\dots .eq\left(1\right)$

for finding the mth term from the end

the first term becomes $a_1=l$

and the common difference becomes $-d$

now mth term from last is given by

$a_m^{\prime }=l+(m-1)(-d)\dots .eq\left(2\right)$

now, we have to find sum of eq(1) and eq(2)

$a_m+a_m^{\prime }=a+(m-1)d+l-(m-1)d$

$a_m+a_m^{\prime }=a+l$