The ${n^{th}}$ term of the series $3 \cdot 8 + 6 \cdot 11 + 9 \cdot 14 + 12 \cdot 17 + \dots$ will be

Let $\sum_{k=1}^{n} a_{k} = \alpha n^{2} + \beta n$. If $a_{10} = 59$ and $a_{6} = 7a_{1}$,then $\alpha + \beta$ is equal to:

If the roots of the equation $x^3+ax^2+bx+c=0$ are in arithmetic progression,then

If $\log _{10} 2, \log _{10} (2^x - 1), \log _{10} (2^x + 3)$ are in $A.P.,$ then :-

$A$ man starts repaying a loan with a first installment of $Rs. 100$. If he increases the installment by $Rs. 5$ every month,what amount will he pay in the $30^{th}$ installment?

$A$ man counts $4500$ currency notes. Let $a_n$ denote the number of notes he counts in the $n^{th}$ minute. If $a_1 = a_2 = \dots = a_{10} = 150$ and $a_{10}, a_{11}, \dots$ form an arithmetic progression with a common difference of $-2$,then how many minutes will it take for him to count all the notes?