Introduction
This STEP support module includes some useful factorisation results, and some work with Fibonacci numbers. Previous assignments can be found here, but you can do this one without having done the others first.
STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").
About this assignment
The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.
The warm up for this assignment introduces the "difference of two cubes" and "sum of two cubes" factorisations, which (like the difference of two squares factorisation) are well worth knowing.
The main STEP question (2010 STEP 2 Question 3) involves Fibonacci numbers and an explicit formula for them (rather than the implicit one $F_{n+1}=F_n+F_{n-1}$).
The final question is a simplified version of a problem from a 1850 publication "The Lady's and Gentleman’s Diary".
For more on Fibonacci numbers, see this article.
You can find the 1850 edition of "The Lady's and Gentleman's Diary" here. The problem posed by Reverend Thomas Kirkman is at the bottom of page 48 of the 1850 edition.
Hints, support and self evaluation
The “Hints and partial solutions for Assignment 14” file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.
Here is a Worked Video Solution to the STEP question from this assignment.