The difference between any two consecutive interior angles of a polygon is $5^{\circ}$. If the smallest angle is $120^{\circ}$,find the number of the sides of the polygon.

Statement-$I$: If the ratio of the sum of $n$ terms of two arithmetic progressions is $(7n + 1) : (4n + 17)$,then the ratio of their $n^{th}$ terms is $7 : 4$.
Statement-$II$: If $S_n = an^2 + bn + c$,then $T_n = S_n - S_{n-1}$.

Given that $n$ $A$.$M$.'s are inserted between two sets of numbers $a, 2b$ and $2a, b$,where $a, b \in R$. Suppose further that the $m^{th}$ mean between these sets of numbers is the same,then the ratio $a:b$ equals

If the roots of $x^3+a x^2+b x+c=0$ are in arithmetic progression with common difference $1$,then

If $\log 2, \log (2^n - 1)$ and $\log (2^n + 3)$ are in $A.P.$,then $n =$

The condition that the roots of $x^3-b x^2+c x-d=0$ are in arithmetic progression is