### Introduction

This is the third of the 25 Foundation modules. We suggest working through the first assignment and second assignment before starting this.

STEP is a challenging examination, and is different in style from A-level, although the mathematical content is the same. STEP questions are longer: they are designed to take around 30 minutes, rather than the typical 10 minutes required for an A-level question.

Do not worry if the STEP problems appear difficult: they are meant to be! However, you should not be daunted. These assignments are designed to help you to develop the skills you need, over time, so that by the time you sit the STEP exam in the summer of Y13 you will feel well-prepared.

### 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 involves the sigma notation, and a proof of the formula for the sum of the terms of a geometric progression. For the last part, use the formula rather than summing the individual terms, and try to do this without using a calculator.

The **main STEP question (2004 STEP 1 Question 2)** introduces the “floor” notation. More information on this and related functions can be found here.

The **final question** involves a linear Diophantine equation, i.e. one of the form ax+by = c where a, b and c are are given integers and we are looking for a solution where x and y are also integers.

### Further reading

More information on Diophantine equations can be found in this article on Plus, the free online mathematics magazine, and in this Wikipedia entry.

You may also enjoy watching this talk by Dr Vicky Neale - 'How to Solve Equations'

This Plus article shows how we can integrate from first principles.

This Numberphile video discusses the "Monkeys and Coconuts" problem. Do not watch it until you have tried the assignment!

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 3”** 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.