Question

Show that sequence of following $n^{th}$ terms ,are not in A.P.
$\frac{n}{n+1}$

Answer 1

The first three terms can be calculated by substituting $n=1,2,3$
Let them be denoted by $a,b,c$ respectively.
$a=\frac{1}{2}$
$b=\frac{2}{3}$
$c=\frac{3}{4}$
$a+c=\frac{5}{4}$
$2b=\frac{4}{3}$
If $a,b,c$ actually were in an A.P., then $a+c$ would be equal to $2b$ which is not the case here. Hence, the given sequence in not an A.P.