Question

if the mth term of an ap is n and nth term of an ap is m find the rth term.

Answer 1

am= n
a+(m-1)d =n. eq.1

an=m
a+(n-1)d= m. eq.2


solving eq1 and eq2
[a+(m-1)d] - [a+(n-1)d] =n - m
a+(m-1)d - a-(n-1)d = n-m
(+a-a=0)
(m-1)d-(n-1)d=n-m
d[(m-1)-(n-1)]=n-m
d[m-1-n+1]=n-m
d[m-n]=n-m
d=(n-m)/m-n
d=-(m-n)/m-n
d=-1

therefore, a+(m-1)d=n (from eq1)
a+(m-1)(-1)=n
a-m+1=n
a= n+m-1
now,rth term= ar=a+(r-1)d
ar= n+m-1+(r-1)(-1)
ar=n+m-1-r+1
ar=n+m-r