|
|
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 |
Using mathematical induction and the definition \( ^nC_r = \frac{n!}{r!(n-r)!} \), prove that
$$ \sum_{r=2}^{n} \; ^rC_2 = \; ^{n+1}C_3 $$for all \( n \in \mathbb{Z}^+ \).
4. | 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.
If you would like space on the right of the question to write out the solution try this Thinning Feature. It will collapse the text into the left half of your screen but large diagrams will remain unchanged.
The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.
The solutions to the questions on this website are only available to those who have a Transum Subscription.
Exam-Style Questions Main Page
To search the entire Transum website use the search box in the grey area below.
Do you have any comments about these exam-style questions? It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Click here to enter your comments.
