If $S_n$ denotes the sum of $n$ terms of an arithmetic progression,then the value of $(S_{2n} - S_n)$ is equal to

If the ${n^{th}}$ terms of two arithmetic progressions $(A.P.)$ are $3n + 8$ and $7n + 15$,then the ratio of their ${12^{th}}$ terms will be:

Let $S_k = \frac{1 + 2 + 3 + .... + k}{k}$. If $S_1^2 + S_2^2 + ....... + S_{10}^2 = \frac{5}{12}A$,then $A$ is equal to

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

If ${a_1}, {a_2}, {a_3}, ..., {a_{24}}$ are in arithmetic progression and ${a_1} + {a_5} + {a_{10}} + {a_{15}} + {a_{20}} + {a_{24}} = 225$,then ${a_1} + {a_2} + {a_3} + ... + {a_{23}} + {a_{24}} = $

The sum of three numbers in a $GP$ is $105$ and their product is $8000$. Find the numbers.