# 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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