If the sum of the $10$ terms of an $A.P.$ is $4$ times the sum of its $5$ terms,then the ratio of the first term to the common difference is:

Which term of the sequence $(-8 + 18i), (-6 + 15i), (-4 + 12i), \dots$ is purely imaginary (in $^{th}$)?

Let $A_1, A_2, \dots, A_{39}$ be $39$ arithmetic means between the numbers $59$ and $159$. Then the mean of $A_{25}, A_{28}, A_{31}$ and $A_{36}$ is equal to:

The income of a person is $Rs. \,3,00,000$ in the first year and he receives an increase of $Rs. \,10,000$ to his income per year for the next $19$ years. Find the total amount he received in $20$ years.

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

Let ${a_1}, {a_2}, \dots, {a_{49}}$ be in $A.P.$ such that $\sum_{k = 0}^{12} {a_{4k + 1}} = 416$ and ${a_9} + {a_{43}} = 66$. If $\sum_{r = 1}^{17} a_r^2 = 140m$,then $m = \dots$