IB Maths AA SL AA 1.3 Proof questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

Difficulty

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Question 1

Paper 1, no calculatorHard6 marks

For $n\in\mathbb Z$ define

S_n=(n+1)^5+(n-1)^5.

A student conjectures that $S_n$ is divisible by $32$ for every odd integer $n$

(a)

Calculate $S_1$ directly.

[1]

(b)

Starting from $S_n$ prove the identity

S_n\equiv 2n^5+20n^3+10n

for all $n\in\mathbb Z$ Use $=$ between successive expressions and $\equiv$ for the final identity.

[2]

(c)

Hence prove the student’s conjecture for every odd integer $n$

[3]

Question 2

Paper 1, no calculatorHard6 marks

Let $m\in\mathbb Z^+$ Figure 1 shows a rectangle with side lengths $m+1$ and $m+2$ divided into four smaller rectangles.

It is claimed that integers $A$ and $B$ can be chosen so that, for every $m\in\mathbb Z^+$

\frac{1}{m+1}+\frac{1}{m+2}+\frac{Am+B}{m^2+3m+2} =\frac{3}{m+1}.

(a)

Using the areas shown in Figure 1, write an identity relating the area of the whole rectangle to the areas of the four smaller rectangles. Use $\equiv$ to denote an identity.

[1]

(b)

If the claim is true, the equality for $m=1$ requires a particular relation between $A$ and $B$ Determine this relation.

[1]

(c)

Determine $A$ and $B$ so that the claim is an identity for every $m\in\mathbb Z^+$ Then, starting with the left-hand side, prove the resulting identity. Use your answer to part (b).

[4]

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