Arithmetic Sequences & Series (DP IB Maths: AI HL)

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2 April 2023

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In an arithmetic sequence, the difference between consecutive terms in the sequence is constant
This constant difference is known as the common difference, d, of the sequence
- For example, 1, 4, 7, 10, … is an arithmetic sequence with the rule ‘start at one and add three to each number’
  - The first term, u₁, is 1
  - The common difference, d, is 3
- An arithmetic sequence can be increasing (positive common difference) or decreasing (negative common difference)
- Each term of an arithmetic sequence is referred to by the letter u with a subscript determining its place in the sequence

$u_{n} = u_{1} + (n - 1) d$

- Where $u_{1}$ is the first term, and $d$ is the common difference
- This is given in the formula booklet, you do not need to know how to derive it
Enter the information you have into the formula and use your GDC to find the value of the term
Sometimes you will be given a term and asked to find the first term or the common difference
- Substitute the information into the formula and solve the equation
  - You could use your GDC for this
Sometimes you will be given two terms and asked to find both the first term and the common difference
- Substitute the information into the formula and set up a system of linear equations
- Solve the simultaneous equations
  - You could use your GDC for this

Simultaneous equations are often needed within arithmetic sequence questions, make sure you are confident solving them with and without the GDC

The fourth term of an arithmetic sequence is 10 and the ninth term is 25, find the first term and the common difference of the sequence.

ai-sl-1-2-2-arit-sequences

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An arithmetic series is the sum of the terms in an arithmetic sequence
- For the arithmetic sequence 1, 4, 7, 10, … the arithmetic series is 1 + 4 + 7 + 10 + …
Use the following formulae to find the sum of the first n terms of the arithmetic series:

$S_{n} = \frac{n}{2} (2 u_{1} + (n - 1) d); S_{n} = \frac{n}{2} (u_{1} + u_{n})$

- - $u_{1}$ is the first term
  - $d$ is the common difference
  - $u_{n}$ is the last term
- Both formulae are given in the formula booklet, you do not need to know how to derive them
You can use whichever formula is more convenient for a given question
- If you know the first term and common difference use the first version
- If you know the first and last term then the second version is easier to use
A question will often give you the sum of a certain number of terms and ask you to find the value of the first term or the common difference
- Substitute the information into the formula and solve the equation
  - You could use your GDC for this

The formulae you need for arithmetic series are in the formula book, you do not need to remember them
- Practice finding the formulae so that you can quickly locate them in the exam

The sum of the first 10 terms of an arithmetic sequence is 630.

Find the common difference, d, of the sequence if the first term is 18.

ai-sl-1-2-2-arit-seriesa

Find the first term of the sequence if the common difference, d, is 11.

ai-sl-1-2-2-arit-seriesb

the (exam) results speak for themselves:

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