The graph of forms a bunch of squares and circles. The area of one of these circles is most nearly which of the following?
Remark:
Find the value of
Suppose a particle moves in a right-angled left spiral on an -grid. That is, it moves a distance in a straight line, stops, makes a right-angled turn to it's "left", travels a distance in a straight line, stops, makes a right angled turn to its "left", travels a distance in a straight line and continues in this fashion forever.
If for and if then find the magnitude of the straight line distance between the particle's starting and finishing points.
For all integers , we define as follows: For all , let Find .
Details and assumptions
As an explicit example, since , , whereas since . Note that .
The floor function denotes the largest integer less than or equal to . For example, .
You might use a scientific calculator for this problem.
If and are roots of the equation then
is equal to?