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The Fibonacci sequence is not the only one that is defined recursively. Learn how to solve combinatorics problems with recursion, and how to turn recurrence relations into closed-form expressions.

What is the area enclosed by a Koch snowflake starting from an equilateral triangle with side length 1?

A. 1

B. \(\frac{1}{2} \)

C. \( \frac{2\sqrt{3}}{5}\)

D. \( 2 \frac{\sqrt{3}}{4}\)

E. Area is infinite

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At a warehouse, robotic arms are responsible for retrieving boxes when an order is received from a customer. In one area of the warehouse, there is a stack of 6 boxes. Each box is a cube, and all the boxes have different sizes. The boxes are stacked from largest to smallest, with the largest box on the bottom.

When doing an inventory check, the warehouse foreman realizes that this stack of boxes is in the wrong spot. There are two empty spots to the right of this stack, and the boxes are supposed to be in the rightmost spot. The foreman programs the robotic arm to move the boxes from where there are now into the new stack. The robotic arm is only able to pick up one box at a time, and for safety reasons, it is never allowed to place any box on top of a smaller box, since it might fall off.

What is the smallest number of moves that the robotic arm must make in order to move the stack of boxes from its present location to where it is supposed to be?

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\[ \begin{eqnarray} A&=&2000\\ \\ B&=&A-999\\ \\ C&=&A+B-998\\ \\ D&=&A+B+C-997\\ & \vdots & \\ Z&=&A+B+C+...+Y-975 \end{eqnarray} \]

How much is \( \frac { Z+1 }{ { 2 }^{ 25 } } ? \)

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