The first term of an arithmetic progression is $10$ and the last term is $50$. If the sum of all its terms is $300$,then the number of terms $n = ...$

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 $-112$.

If $n$ is a natural number,then $\left( \frac{n+1}{2} \right)^n \ge n!$ is true when

Let $a_1, a_2, a_3, \ldots, a_{11}$ be real numbers satisfying $a_1=15$,$27-2a_2 > 0$,and $a_k = 2a_{k-1} - a_{k-2}$ for $k = 3, 4, \ldots, 11$. If $\frac{a_1^2 + a_2^2 + \ldots + a_{11}^2}{11} = 90$,then the value of $\frac{a_1 + a_2 + \ldots + a_{11}}{11}$ is equal to

Let the six numbers $a_1, a_2, a_3, a_4, a_5, a_6$ be in $A.P.$ and $a_1+a_3=10$. If the mean of these six numbers is $\frac{19}{2}$ and their variance is $\sigma^2$,then $8 \sigma^2$ is equal to

Suppose $a_1, a_2, a_3, \ldots, a_{2012}$ are integers arranged on a circle. Each number is equal to the average of its two adjacent numbers. If the sum of all even-indexed numbers is $3018$,what is the sum of all numbers?