One fine summer morning not so long ago, I woke up with a start to the stark realization that I had no idea how to derive the formula for the sum of powers of natural numbers. I am talking about a closed-form expression for the series (for non-negative integer \(p\)) \[ \sum_{i=1}^n {i^p} = 1 + {2^p} + {3^p} + \dots + {n^p} \]
Last week, for pedagogical reasons, I had to solve the following problem:
In a bag there are \(4\) blue and \(5\) red balls. We randomly take \(3\) balls from the bag without replacement. Let the random variable \(X\) denote the number of blue balls we get. Find out the expectation and variance of \(X\).
Years back, I encountered the following problem in a book on Number Theory: Prove that the sum \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}\) is never an integer after \(n > 1\). In other words, in the sequence \(1, 1+\frac{1}{2}, 1+\frac{1}{2}+\frac{1}{3}, \dots\), there is only one integer, and that is the first number \(1\).
A famous problem in probability is this: Suppose you randomly1 break a stick in two places. You will get three smaller pieces of stick. What is the probability that you can form a triangle using these pieces?
One interesting thing about this problem is that depending on how you visualize breaking the stick, you get different probabilities.