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IB MATH AA HL, Paper 3, November, 2022, TZ0, Solved Past Paper

Master the 2022 IB November for Paper 3 Mathematics AA HL with examiner tailored solutions and comments for TZ0

Question 1 [Explained]

In this question, we will explore series represented by the sum $\sum_{i = 1}^{n} i^{q} = 1^{q} + 2^{q} + 3^{q} + \dots + n^{q}$ where (n) and (q) are positive integers. We aim to derive polynomial expressions in terms of (n) that can calculate the total of such series for any given power (q).

We will employ different mathematical strategies to develop these formulas. This exploration is essential for understanding how to generalize the summation of powers, which is a foundational concept in higher-level mathematics, particularly useful in fields such as engineering and physics.

Question 1 [a] [Explanation]

Consider the scenario where (q = 1), making the series in question an arithmetic series. Under this condition, your task is to demonstrate the following summation property:

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

This problem essentially asks you to verify that the sum of the reciprocals of the first (n) natural numbers equals half the product of (n) and (n + 1)). This is a classic result involving a straightforward arithmetic series when the common difference of the sequence is 1.

Video Solution by an IB Examiner - Coming soon

Question 1 [b] [Explanation]

Consider a scenario where the variable (q) is set to 2.

The table presented below outlines two mathematical concepts: the square of (n), denoted as (n²), and the cumulative sum of (i²) from (i = 1) to (n), symbolized as $(\sum_{i = 1}^{n} i^{2})$ . This table provides these values for (n) values of 1, 2, and 3.

n	n²	$\sum_{i = 1}^{n} i^{2}$
1	1	1
2	4	5
3	9	P

Question 1 [b] [i] [Explanation]

Please provide the numerical value for the variable (p).

This question asks for a specific calculation related to the variable (p), often found in mathematical or scientific contexts where determining such values is essential for further analysis or problem solving.

Question 1 [b] [ii] [Explanation]

The sum of the squares of the first (n) natural numbers can be represented by a cubic polynomial, which is a type of algebraic equation consisting of three terms. Specifically, this sum can be described using the formula:

$(\sum_{i = 1}^{n} i^{2} = a_{1} n + a_{2} n^{2} + a_{3} n^{3})$ , where (a₁), (a₂), and (a₃) are coefficients that belong to the set of rational numbers $((Q^{+}))$ .

To find the values of these coefficients, you need to create and solve a system of three linear equations. These equations will allow you to express the coefficients (a₁), (a₂), and (a₃) in terms of (n), thus defining how the polynomial changes as (n) varies.