# July, 2020

## Ask Uncle Colin: A Sum With An Unknown

Dear Uncle Colin, My textbook gives me an arithmetic sequence that starts $3 + 8 + 13 + \dots$, and asks me to find where the sum is 1575. I’ve got it down to $\frac{-1 \pm \sqrt{63,001}}{10}$, but I don’t know how to work out that square root without a

## The Mathematical Ninja and Logs Base 2

The student’s shoulder twitched slightly as he said “So I need to work out $\log_2(10)$…” and the crash of the cane against the table reminded him that the calculator was off-limits. “I think you can estimate that yourself,” said the Mathematical Ninja. “Uh… ok. There’s a change of base formula,

Dear Uncle Colin, How would you go about factorising $6x^2 - xy - y^2 + 7x - y + 2$? - Argh! Getting Nowhere. Expression Simplification Impossible. Hi, AGNESI, and thanks for your message! I have, historically, not been a fan of these. However, I’ve recently come across a method

## Summing with Generating Functions

A nice challenge puzzle via Reddit: Find $\sum_{n=1}^{\infty} \frac{n2^n}{(n+2)!}$ There was a video attached to it that I didn’t watch, something about telescoping sums, but the moment I saw this, I thought: generating functions! Why would I think such a thing? The thing that jumped out at me was the

## Ask Uncle Colin: Why does the $ac$ method work?

Dear Uncle Colin, A while back, you shared an easy way to factorise nasty quadratics. Why does it work? Dutifully Indulging Students’ Curiousity & Reasoning In Maths. I’m Not A Nasty Teacher! Hi, DISCRIMINANT, and thanks for your message! Let’s start by recapping the method in the post, go through

## Captain Holt’s Seesaw

When Jake’s father (Bradley Whitford) comes to town, Jake is excited to see him, but Charles is wary of his intentions; Holt challenges Amy, Terry, Gina and Rosa with a brain teaser in exchange for Beyonce tickets. Brooklyn 99, S02 E18, Captain Peralta “Surely it isn’t that hard?” I have

## Ask Uncle Colin: Dividing Decimals

Dear Uncle Colin, I’m trying to work out $0.03 \div 0.1$. The answer is apparently 0.3, but I don’t understand why a division would make something bigger! Division Isn’t Very Intuitive, Decimals Especially Hi, DIVIDE, and thanks for your message! There are several ways to tackle this. I’ll start with

## Dictionary of Mathematical Eponymy: The Sieve of Sundaram

I am not a number theorist. I mean… well scratch that. I am not an especially knowledgeable number theorist1 but I do enjoy number theory when it’s around my level. The Sieve of Sundaram is about my level. What is the Sieve of Sundaram? OK, let’s start with the Sieve

## Ask Uncle Colin: The Area Between Two Curves

Dear Uncle Colin, I need to find the area between the graphs of $y=\sin(2x)$ and $y=\cos^2(x)$ in the interval $0 \le x \lt 2\pi$. I’ve found four solutions, but I think that means I need to do five separate integrals! Is there an easier way? Trigonometric Expressions: Double Integration Of