For a natural number $n$,let $a_{n} = 19^{n} - 12^{n}$. Then,the value of $\frac{31 a_{9} - a_{10}}{57 a_{8}}$ is

If $[x]$ represents the greatest integer not greater than $x$,then $\left[\left(1+\frac{1}{100000}\right)^{100000}\right]=$

If $x$ is the arithmetic mean and $y, z$ are the two geometric means between two positive numbers,then $\frac{y^3 + z^3}{xyz} = \dots..$

If $a, b, c$ are three consecutive terms of an $A.P.$ and $x, y, z$ are three consecutive terms of a $G.P.$,then the value of $x^{b-c} \cdot y^{c-a} \cdot z^{a-b}$ is:

If $a, b, c$ are in $G.P.$ and $\log a - \log 2b, \log 2b - \log 3c$ and $\log 3c - \log a$ are in $A.P.$,then $a, b, c$ are the lengths of the sides of a triangle which is

Write the first five terms of the following sequence and obtain the corresponding series:
$a_{1} = a_{2} = 2, a_{n} = a_{n-1} - 1, n > 2$