IB Maths AA HL 1.15 Advanced proof Question Bank

Practise IB Mathematics HL 1.15 by applying advanced proof methods to exam-style questions.

[Maximum number: 6]

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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AHL 1.15 (HL)—Advanced proof question 1