IB Maths AA HL Ahl 1 15 Hl Advanced Proof Questions

Practise proving statements by induction or contradiction, and disproving universal claims with a counterexample that identifies the violated condition precisely.

Exam points

Set up a mathematical induction proof with a valid base case and an inductive step, then state the conclusion for every permitted integer n.
Use contradiction to combine the assumptions with parity, divisibility, inequalities or integer conditions and derive an explicit impossibility.
Disprove a universal claim with a valid counterexample and explain precisely which asserted condition it violates.

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Study AHL 1.15 (HL)—Advanced proof conceptsOpen the corresponding Concept lesson for explanations, examples and exam connections.

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In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

Prove by mathematical induction that $f_{q}(x)=\sum_{i=1}^{n} i^{q} x^{i}, q \in \mathbb{Z}^{+}$ .

consider q=1

f_{1}(x)=x+2 x^{2}+\ldots+n x^{n} \text { (reference to part (c)) and } f_{1}(x)=\sum_{i=1}^{n} i x^{i}

assume true for $q=k,\left(f_{k}(x)=\sum_{i=1}^{n} i^{k} x^{i}\right)$

Note: Do not award M1 for statements such as "let q=k " or " q=k is true".
Subsequent marks after this M1 are independent of this mark and can be awarded.
consider q=k+1

\begin{aligned} & f_{k+1}(x)=x f_{k}^{\prime}(x) \\ & =x \sum_{i=1}^{n} i^{k+1} x^{i-1} \quad \text { OR } \quad x\left(1+2^{k+1} x+3^{k+1} x^{2}+\ldots+n^{k+1} x^{n-1}\right) \end{aligned}

Note: Award the above M1 if $f_{k+1}(x)=x \sum_{i=1}^{n} i^{k+1} x^{i-1}$ or $x f_{k}^{\prime}(x)=x \sum_{i=1}^{n} i^{k+1} x^{i-1}$ (or equivalent) is stated.

=\sum_{i=1}^{n} i^{k+1} x^{i} \text { OR } x+2^{k+1} x^{2}+3^{k+1} x^{3}+\ldots+n^{k+1} x^{n}

since true for q=1 and true for q=k+1 if true for q=k, hence true for all $q\left(\in \mathbb{Z}^{+}\right)$

Note: To obtain the final R1, three of the previous five marks must have been awarded.

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IB Maths AA HL Ahl 1 15 Hl Advanced Proof Questions question 1