Find the common difference of an $AP$ whose first term is $5$ and the sum of its first four terms is half the sum of the next four terms.

The ${17}^{th}$ term of$AP$ is $5$ more than twice its$8^{th}$ term. If the ${11}^{th}$ term of the $AP$ is $43$, find its $n^{th}$ term.

The ${24}^{th }$term of an $AP$ is twice its ${10}^{th}$ term. Show that its ${72}^{nd}$ term is $4$ times its ${15}^{th }$term.

The ${19}^{th}$ term of an $AP$ is equal to $3$ times its $6^{th}$ term. If its $9^{th}$ term is $19$, find the $AP$.

If the $p^{th}$term of an $AP$ is  $q$and its $q^{th}$ term is $p$ then show that its $(p + q{)}^{th}$term is zero.

The first and last terms of an $AP$ are $a$ and $l$ respectively. Show that the sum of the $n^{th}$ term from the beginning and the $n^{th}$ term from the end is$( a + l )$.

How many two-digit numbers are divisible by $6$?