Question

In an A.P, $m^{th}$ term is $n$ and $n^{th}$ term is $m$, show that its $r^{th}$ term is $(m+n-r)$.

Answer 1

Given :- am = n

an = m

Solution :-

am = n

a + ( m - 1 )d = n .... ( i )

an = m

a + ( n - 1 )d = m.... ( ii )

Sub ii from i

a + ( m - 1 )d - a - ( n - 1 )d = n - m

md - d - nd + d = n - m

md - nd = n - m

d ( m - n ) = n - m

d = - ( m - n ) / ( m - n )

d = - 1

Putting value of d in ( i )

n = a + ( m - 1 )d

n = a + ( m - 1 ) × ( - 1 )

n = a - m + 1

n + m - 1 = a

ar = a + ( r - 1 ) × ( - 1 )

ar = n + m - 1 - r + 1

ar = n + m - r

Hence proved !!