If $u_{0}=8, u_{1}=3, u_{2}=12, u_{3}=51$,then the value of $\Delta^{3} u_{0}$ is

If the square root of $a^{\frac{1}{a}} \cdot (2a)^{\frac{1}{2a}} \cdot (4a)^{\frac{1}{4a}} \cdot (8a)^{\frac{1}{8a}} \cdots \infty$ is $\frac{8}{27}$,then the value of $a$ is:

If $(10)^9 + 2(11)^1(10)^8 + 3(11)^2(10)^7 + \dots + 10(11)^9 = k(10)^9$,then $k$ is equal to:

If $x, y, z$ are three positive numbers in $G.P.$,then $\frac{1 + \ln x}{2}, \frac{1 + \ln y}{4}, \frac{1 + \ln z}{8}$ are in

If $7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \dots \infty$,then the value of $\alpha$ is:

Let $p(x)=a_0+a_1 x+\ldots+a_n x^n$. If $p(-2)=-15, p(-1)=1, p(0)=7, p(1)=9, p(2)=13$ and $p(3)=25$,then the smallest possible value of $n$ is