## Exam-Style Questions.## Problems adapted from questions set for previous Mathematics exams. |

## 1. | IB Analysis and Approaches |

Use mathematical induction to show that:

$$\sum_{r=1}^n \dfrac{1}{r(r+1)} = \dfrac{n}{n+1} $$for all \(n \in \mathbb{Z}^+\).

## 2. | IB Analysis and Approaches |

Consider the sum of the squares of any two consecutive odd integers.

(a) Show that \((2n + 1)^2 + (2n + 3)^2 = 8n^2 +16n + 10\) , where \(n \in \mathbb{Z} \)

(b) Hence, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.

## 3. | IB Analysis and Approaches |

Prove that the integers \(a\) and \(b\) cannot both be odd if \(a^2+b^2\) is exactly divisible by 4.

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