With a second UK lockdown looming, many of us have once again found ourselves back inside looking for entertainment. Whether it’s home workouts, computer games, baking banana bread or a mixture of the three, we all have different preferences for passing the time. If you’ve found yourself here, it’s likely that this may also include some equations. Whether you’re a student, or enjoy thinking and reading all things mathematical for recreation, why not further improve your understanding of some of the most common formula by challenging yourself with finding their proofs. A satisfying but rewarding task, here are three we’ve chosen: see if you can solve them!

1) The quadratic formula

The solutions to the quadratic equation 𝑎𝑥2+𝑏𝑥+𝑐=0

are given by 𝑥= −𝑏±√𝑏2−4𝑎𝑐2𝑎

A well-known formula that is used throughout secondary school mathematics and beyond. However, its derivation is rarely covered. A solvable problem for any students who have studied quadratic equations, why not give it a go? Don’t be intimidated by the algebra and perhaps think about completing the square…

2) Sum of the first 𝑛 terms of an arithmetic sequence

An arithmetic sequence is one where, from term to term, the values increase by a constant amount. For example, 2,6,10,14,18,..

To find the sum of the first ‘𝑛’ of these terms, there is a very handy formula: 𝑆𝑛=𝑛(𝑎1+𝑎𝑛)2

For an experienced mathematician, deriving this formula is quite a simple task, but for those still at school or just looking to refresh it’s a good little brain trainer!

3) Bayes’ theorem

𝑃(𝐴|𝐵)=𝑃(𝐵|𝐴)𝑃(𝐴)𝑃(𝐵)

A fundamental theorem for probability, Bayes’ theorem describes the relationship of the conditional probability between two events. The proof is simple once you know how! See if you can spot it (hint: consider the definitions of each component).

Maths in the media…

- For the Pi lovers https://twitter.com/3blue1brown/status/1305938925098209280

 - And another for Lord of the Rings fans.. https://twitter.com/fermatslibrary/status/1307327128611041280

 - After all the ‘back to school’ talk, there’s also the small matter of thousands of new students starting university in a global pandemic.. some interesting statistics on how this might unfold

https://plus.maths.org/content/going-back-uni-during-pandemic

 - A new way to visualise General RelativityWhat would Einstein think… https://www.youtube.com/watch?v=7hXdlNZJ_BY