Let $a_1, a_2, a_3, \dots$ be an $A.P.$ with $a_6 = 2$. Then the common difference of this $A.P.$,which maximizes the product $a_1 a_4 a_5$,is

Let the sequence $a_{n}$ be defined as follows:
$a_{1} = 1, a_{n} = a_{n-1} + 2$ for $n \ge 2$
Find the first five terms and write the corresponding series.

If the $n^{th}$ term of an arithmetic progression is $\frac{(2n + 1)}{3}$,what is the sum of its first $19$ terms?

Let $A, G, H$ and $S$ respectively denote the arithmetic mean,geometric mean,harmonic mean and the sum of the numbers $a_1, a_2, a_3, \ldots, a_n$. Then the value of $x$ at which the function $f(x)=\sum_{k=1}^n(x-a_k)^2$ has a minimum is

If $x, y, z \in \mathbb{R}^+$ such that $x + y + z = 4$,then the maximum possible value of $xyz^2$ is

Let the sequence $a_1, a_2, a_3, \dots, a_{2n}$ form an $A.P.$ Then $a_1^2 - a_2^2 + a_3^2 - a_4^2 + \dots + a_{2n - 1}^2 - a_{2n}^2 = $