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Let f(k) be the sum of the first k odd natural numbers; that is, f(k)=1+2+…+(2k?1).  Compute the values f(k) for several values.  Can you see a pattern?  Can you prove the pattern?
Which positive integers can be written as the sum of two squares of (nonnegative) integers?  For example, 4=02+22, 5=12+22, and 8=22+22.  However, 3 cannot be written as the sum of two squares.  Determine which of the first 100 natural numbers can be written as the sum of two squares.  What patterns do you observe? How could you gain more evidence to support your conjectures?

Please help me write a sage function for the following problem:

Suppose a gambler starts with R1000. He repeatedly plays a game where the initial bet is R100. The gambler has a 60% chance of losing the game (in which case he gets nothing), but a 40% chance of winning and doubling his money (ie, he gets back R200). If the gambler loses all of his money, he is ruined and stops playing. If the gambler reaches R2000, he also stops playing. Otherwise, the gambler keeps playing. What is the probability that the gambler stops with R2000? What is the probability he is ruined?

Write a simulation to empirically determine the probabilities. Run your simulation 10000 times. What are the probabilities for ruin and winning?

Q: Let A be a square 2×2 matrix. The linear transformation x?Ax maps R2 to R2. Create an interaction that shows the effect of applying A to the unit square. Draw two images: the plane before the linear transformation (showing the unit square and colored basis vectors) and the plane after the linear transformation (showing the image of the unit square under A and the image of the basis vectors). It may be useful to use the "graphics array" to show two plots next to each other. In the interact, allow the user to enter 4 values that are the entries of the 2×2 matrix A.

How can I go about this?