### Introduction

This STEP support module includes some work on sequences, some notation which you may not have seen before and a problem about crossing the desert. 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 practises algebraic manipulation and then introduces the product notation (very similar to sigma notation for summations!)

The **main STEP question (2006 STEP 2 Question 1)** is about sequences, and whether they are periodic or converge.

The **final question** is a problem about crossing the desert if your car can only hold a certain amount of petrol.

For more on sequences, including discussion of the "limits inferior and superior", see this Plus maths article.

You may like to investigate the properties of the **logistic map** which models population growth and is defined by $x_{n+1}=rx_n(1-x_n)$ and $0 \le r \le 4$. Use a spreadsheet to see what happens for different values of $r$. There are some nice animations on this Wikipedia page.

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 15”** 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.