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Chapter 1: Warmup

  1. What is the intuition for this infinite series being equal to 1? $$ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \dots$$

  2. Fact: $$ \frac{1}{1-x} = 1+x+x^2+x^3+\dots $$. Is this always true?

  3. Note that $$ e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!}\dots $$. Why is this number important? Can you notice that this number is obviously greater than 1? Can you show that it is actually less than 3?

  4. Note that $$ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!}\dots $$. What is the significance of this?

  5. Understand Binomial Theorem and show that $$ \left( 1+\frac{1}{n}\right)^n = e) $$

  6. If you keep compounding your money, when will it get doubled? (Hint: $$ \left( 1+\frac{1}{n}\right)^n = e) $$)

  7. If you keep spending your money by $$\epsilon$$ proportion, when will your money get to less than a rupee?

  8. If you consider a 1000 meter long stick and randomly break it at some point, what do you expect the length to be, of the smaller part and the bigger part?

  9. A bus conductor charges 1000 rupees per seat. The bus can accommodate 100 people. If $$k$$ number of seats are empty, the conductor levies a penalty of $$10k$$ on each passenger. For example, if 5 seats are empty, the conductor will charge 1050 for the 95 passengers who are present. What is the number of passengers for which the conductor earns the maximum money?

  10. Understand the Monty Hall problem.

  11. What is the importance of a matrix? Where do we use them?

  12. What is the intuitive idea of an inverse of a matrix?

  13. In the following code, what is the probability that $$a$$, $$b$$ and $$c$$ form the sides of a triangle?

    import random
a=random.random()
b=random.random()
c=random.random()

  14. Show that $$sqrt(2)$$ is irrational.

  15. Show that prime numbers are infinite.

  16. You know the old school way of adding, subtracting, multiplying and dividing 2 big numbers. Can you provide a proof for these methods?

  17. Find the GCD of two big numbers using only a calculator. What method helps you calculate the GCD fast? How do you know that the method works? Can you figure out the number of steps it takes to arrive at a solution?

  18. Describe Birthday Paradox. Write a piece of python code to figure out the probability of you finding same birthdays in a class of $$n$$ people. Plot the probability on the $$Y$$ axis and the number of people in the class $$n$$ in the $$X$$ axis. How does the curve look like?

  19. After programmatically understanding the birthday paradox, can you understand it mathematically?

  20. When you distribute $$n$$ chocolates to $$n$$ people uniformly at random, many of them may end up getting more than one chocolate. How many people, on an average, end up getting no chocolates at all.