Question

Which of the following sequences are in an A.P.?
$\frac{2}{3},\frac{4}{5},\frac{6}{7},....$

Answer 1

Let $t_n,nϵN$ be the $n^{th}$ term of the given sequence.
Therefore, $t_1=\frac{2}{3},t_2=\frac{4}{5},t_3=\frac{6}{7}$
So $t_2-t_1=\frac{4}{5}-\frac{2}{3}=\frac{12-10}{15}=\frac{2}{15}$
$t_3-t_2=\frac{6}{7}-\frac{4}{5}=\frac{30-28}{35}=\frac{2}{35}$
Since $t_2-t_1\ne t_3-t_2$, the given sequence is not an A.P.