The sums of $n$ terms of two arithmetic series are in the ratio $(2n + 3) : (6n + 5)$. Find the ratio of their $13^{th}$ terms.

If $|x| < 1, |y| < 1$ and $x \neq y,$ then the sum to infinity of the following series $(x+y)+(x^{2}+xy+y^{2})+(x^{3}+x^{2}y+xy^{2}+y^{3})+\ldots$ is:

Let ${a_n}$ be the ${n^{th}}$ term of the $G$.$P$. of positive numbers. Let $\sum\limits_{n = 1}^{100} {{a_{2n}}} = \alpha $ and $\sum\limits_{n = 1}^{100} {{a_{2n - 1}}} = \beta $,such that $\alpha \ne \beta $,then the common ratio is

Write the seventh term of the following $GP$: $21a, 84a^3, 336a^5, 1344a^7, \dots$

Find the sum of the numbers of three digits divisible by $7$.

The interior angles of a polygon are in $A.P.$ If the smallest angle is $120^{\circ}$ and the common difference is $5^{\circ}$,then the number of sides is