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}} = $

$A$ person is to count $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \ldots = a_{10} = 150$ and $a_{10}, a_{11}, \ldots$ are in an $A.P.$ with common difference $-2$,then the time taken by him to count all notes is ............... minutes.

If $A = 1 + r^z + r^{2z} + r^{3z} + .......\infty$,then the value of $r$ will be

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

The $n^{th}$ term of the series $\frac{1}{1} + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + \dots$ will be

If $a_1, a_2, ..., a_n$ are positive real numbers whose product is a fixed number $c$,then the minimum value of $a_1 + a_2 + ... + a_{n-1} + 2a_n$ is