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).

Answer 1

Solution 1 :

Given :
First term of A.P is a
nth term =l
Let d be the common difference
nth term=a+(n-1)d
l= a+(n-1)d
d=(l-a)/n-1 ------------1

Mth term from the beginning
=a+(m-1)d=a+(m-1)(l-a/n-1 ) ----------------2

Mth term from the end we get:
where we consider first term as l and the common difference=-d
=l+(m-1)(-d)
=l+(m-1)x(-(l-a)/n-1)
=I-(m-1)(l-a)/n-1 --------------3

thus sum of mth term from beginning to end is
=a+(m-1)(l-a/n-1 ) +I-(m-1)(l-a)/n-1
=a+l
Hence proved

solution 2 :

Here, mth term from last = (n-m+1)th term from begining.

Now Sum of mth term from begining and mth term from last
= a(m)th term+ a(n-m+1)th term
= a + (m - 1)d + a + (n - m + 1 - 1)d
= 2a + (m - 1 +n - m)d
= a + a + (n - 1)d
= a + an
= a + L