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[Maximum number: 2]

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.

(a)

Show that $\sum_{i=1}^{n} i=\frac{1}{2} n(n+1)$ .

Consider the case when q=2.

[ 1 ]

EITHER
$S_{n}=\frac{n}{2}(2 \times 1+(n-1) \times 1)$

OR
$u_{1}=1$ and either $u_{n}=n$ or d=1 stated explicitly

OR
$1+2+\ldots+n$ (or equivalent) stated explicitly

THEN
$S_{n}=\frac{n}{2}(1+n)$

Note: Award A0 for a numerical verification.

(b)

Using sigma notation, write down an expression for $f_{q}(1)$ .

[ 1 ]

$f_{q}(1)=1^{q}+2^{q}+3^{q}+\ldots+n^{q}$

=\sum_{i=1}^{n} i^{q}\left(=\sum_{i=1}^{n} 1^{i} i^{q}\right)

IB Maths AA HL Sl 1 2 Arithmetic Sequences and Series Topic Practice

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IB Maths AA HL Sl 1 2 Arithmetic Sequences and Series Topic Practice

Learn this topic

Question 1

Question (a)

Question (b)