Find the sum of all integers between $50$ and $500$ which are divisible by $7.$

Find the $9^{th}$ term of the $H.P. 6, 4, 3, \ldots$

If the $7^{th}$ term of a harmonic progression is $8$ and the $8^{th}$ term is $7$,then its $15^{th}$ term is

How many terms of the $A.P.$ $1, 4, 7, \ldots$ are needed to give the sum $715$?

If $x = \sum_{n = 0}^\infty a^n$,$y = \sum_{n = 0}^\infty b^n$,and $z = \sum_{n = 0}^\infty (ab)^n$,where $a, b < 1$,then:

If $|\alpha| < 1$ and $|\beta| < 1$,where $s_1 = 1 - \alpha + \alpha^2 - \alpha^3 + \dots \infty$ and $s_2 = 1 - \beta + \beta^2 - \beta^3 + \dots \infty$,then $1 - \alpha\beta + \alpha^2\beta^2 - \alpha^3\beta^3 + \dots \infty$ equals: