Consider four consecutive integers in an increasing arithmetic progression. One of these integers is equal to the sum of the squares of the other three. What is the sum of all the numbers?

If $3^{2 \sin 2 \alpha - 1}$,$14$,and $3^{4 - 2 \sin 2 \alpha}$ are the first three terms of an $A.P.$ for some $\alpha$,then the sixth term of this $A.P.$ is

In an $A.P.$,the first term is $2$ and the sum of the first five terms is one-fourth of the next five terms. Show that the $20^{th}$ term is $-112$.

The houses on one side of a road are numbered using consecutive even numbers. The sum of the numbers of all the houses in that row is $170$. If there are at least $6$ houses in that row and $a$ is the number of the sixth house,then:

The sum of the numbers between $100$ and $1000$ which are divisible by $9$ is:

If the sum of the first $2n$ terms of the arithmetic progression $2, 5, 8, \dots$ is equal to the sum of the first $n$ terms of the arithmetic progression $57, 59, 61, \dots$,then $n = \dots$