IB Mathematics AA SL Arithmetic Sequences & Series Study Notes

LEARNING OBJECTIVE

Arithmetic Sequences & Series

Key Concepts:

Arithmetic Sequences & Series
Sigma Notation

Arithmetic Sequences and Series

Definition (Sequence): An arithmetic sequence is a list of numbers in which the difference between any two consecutive terms is constant. This constant is called the common difference, denoted by $ d $.

General form:

$ a,\, a + d,\, a + 2d,\, a + 3d,\, \dots $

$ a $: the first term
$ d $: the common difference
$ n $: the number of terms

General term (nth term):

$ u_n = a + (n – 1)d $

Definition (Series): The sum of the terms of an arithmetic sequence is called an arithmetic series.

Sum of the first $ n $ terms:

$ S_n = \frac{n}{2} \left[ 2a + (n – 1)d \right] $ or $ S_n = \frac{n}{2} (a + l) $

$ l $: the last term, $ l = a + (n – 1)d $

Properties:

Each term increases (or decreases) by a constant amount.
Graphing an arithmetic sequence gives a straight line.
The mean of the first and last terms is the mean of the entire series.

Example

The first term of an arithmetic sequence is $ a = 4 $, and the common difference is $ d = 3 $.

Find:

The 10th term
The sum of the first 10 terms

▶️ Answer/Explanation

Use the nth term formula:

$ u_{10} = a + (10 – 1)d = 4 + 9 \cdot 3 = 4 + 27 = \rm{31} $

Use the sum formula:

$ S_{10} = \frac{10}{2} [2a + (10 – 1)d] = 5 [2(4) + 9 \cdot 3] = 5 [8 + 27] = 5 \cdot 35 = \rm{175} $

Use of Sigma Notation for Arithmetic Sequences

Definition: Sigma notation is a shorthand way to write the sum of a sequence using the Greek letter $ \sum $ (sigma).

General form:

$ \sum_{k=1}^{n} u_k $, where $ u_k $ is the general term of the sequence

For arithmetic sequences, we typically use:

$ u_k = a + (k – 1)d $

$ a $: first term
$ d $: common difference
$ k $: index of summation
$ n $: number of terms

Sum of arithmetic sequence using sigma notation:

$ \sum_{k=1}^{n} \left[ a + (k – 1)d \right] = \frac{n}{2} [2a + (n – 1)d] $

Key Points:

Make sure the general term inside the sigma is linear (arithmetic form).
Adjust the lower and upper limits accordingly when re-indexing.
The expression inside the sigma should match the general term formula.

Example

Use sigma notation to evaluate the sum:

$ \sum_{k=1}^{5} \left( 3 + 2(k – 1) \right) $

▶️ Answer/Explanation

Identify the sequence:

The general term is $ u_k = 3 + 2(k – 1) $, which simplifies to:

$ u_k = 2k + 1 $ — an arithmetic sequence with $ a = 3 $, $ d = 2 $

Expand the terms:

$ u_1 = 3, u_2 = 5, u_3 = 7, u_4 = 9, u_5 = 11 $

Find the sum:

$ \sum_{k=1}^{5} u_k = 3 + 5 + 7 + 9 + 11 = \rm{35} $

Or use formula:

$ S_n = \frac{n}{2} [2a + (n – 1)d] = \frac{5}{2} [6 + 8] = \frac{5}{2} \cdot 14 = \rm{35} $

Applications of Arithmetic Sequences: Simple Interest

Simple Interest is a real-life example of an arithmetic sequence. When money is invested at simple interest, the amount increases by a fixed amount each year — making it a linear (arithmetic) growth.

Formula for Simple Interest:

$ A_n = P + nI $

$ A_n $: total amount after $ n $ years
$ P $: initial principal
$ I $: annual interest (a constant amount)

This forms an arithmetic sequence with first term $ A_0 = P $ and common difference $ d = I $.

Note: You can also write the total interest earned as:

$ \text{Interest} = P \cdot r \cdot n $

where $ r $ is the annual interest rate (in decimal form).

Example

John invests $1,000 at a simple interest rate of 6% per year. Find the total amount after 5 years.

▶️ Answer/Explanation

$ P = 1000 $
$ r = 6\% = 0.06 $
$ n = 5 $

Calculate yearly interest

$ I = P \cdot r = 1000 \cdot 0.06 = 60 $

Use arithmetic sequence formula:

$ A_n = P + nI = 1000 + 5 \cdot 60 = 1000 + 300 = \rm{1300} $

John will have $1300 after 5 years with simple interest.

Using Technology to Generate and Display Sequences

Example 1: Spreadsheet (e.g., Microsoft Excel or Google Sheets)

Create an arithmetic sequence starting from 2 with a common difference of 3.
In cell A1, enter =2.
In cell A2, enter =A1+3.
Drag down to fill the column and generate the sequence.

Example 2: GDC (TI-84 Plus)

Press MODE → Choose SEQUENCE mode.
Press Y= and enter a recursive formula like u(n)=u(n-1)+2, with u(1)=1.
Press GRAPH or 2nd → TABLE to display the sequence values.

Example 3: Graphing Software (e.g., Desmos)

Use Desmos to graph a sequence using its list and sequence functions.
Enter: f(n) = 2n + 1 where n = 1...20.
Visualize how the sequence behaves and identify patterns.

Approximate Arithmetic Sequences

In real-life situations, data may not follow a perfectly arithmetic pattern. However, if the differences between terms are approximately constant, we can still apply arithmetic models to analyze and make predictions.

Identify an approximate common difference and use it to estimate future values.

Steps:

Collect the data values over equal intervals (e.g., time).
Calculate the differences between successive values.
If the differences are nearly constant, assume an approximate common difference $ d $.
Use the formula $ a_n = a_1 + (n-1)d $ to estimate future terms.

Note: Approximation is necessary in many real-world settings such as finance, population growth, or temperature data, where trends are not perfectly linear.

Example

A college recorded its student enrollment over 5 years as follows:

Year 1: 980 students
Year 2: 1,020 students
Year 3: 1,050 students
Year 4: 1,080 students
Year 5: 1,115 students

Estimate the number of students expected in Year 6 using an approximate arithmetic model.

▶️ Answer/Explanation

Solution:

yearly differences:

- Year 2 – Year 1: $ 1020 – 980 = 40 $
- Year 3 – Year 2: $ 1050 – 1020 = 30 $
- Year 4 – Year 3: $ 1080 – 1050 = 30 $
- Year 5 – Year 4: $ 1115 – 1080 = 35 $

Average difference $ d \approx \frac{40 + 30 + 30 + 35}{4} = 33.75 $

arithmetic sequence formula:
$ a_6 = a_5 + d = 1115 + 33.75 = \rm{1148.75} $

Estimated student population in Year 6: approx. $ \rm{1149} $ students

IB Mathematics AA SL Arithmetic Sequences & Series Study Notes