The maximum sum of the series $20 + 19\frac{1}{3} + 18\frac{2}{3} + \dots$ is

Consider four consecutive integers in an increasing arithmetic progression. One of these integers is equal to the sum of the squares of the other three. What is the sum of all the numbers?

If the roots of the equation $4x^3 - 12x^2 + 11x + k = 0$ are in arithmetic progression,then $k$ is equal to

If $S_k$ denotes the sum of the first $k$ terms of an arithmetic progression whose first term and common difference are $a$ and $d$ respectively,then $S_{kn}/S_n$ is independent of $n$ if

The sum of the common terms of the following three arithmetic progressions:
$3, 7, 11, 15, \ldots, 399$
$2, 5, 8, 11, \ldots, 359$ and
$2, 7, 12, 17, \ldots, 197$,is equal to $................$.

Let $9 < x_1 < x_2 < \ldots < x_7$ be in an $A.P.$ with common difference $d$. If the standard deviation of $x_1, x_2, \ldots, x_7$ is $4$ and the mean is $\overline{x}$,then $\overline{x} + x_6$ is equal to: