SL 1.2 Model arithmetic sequences and series | International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Number and Algebra

From repeated change to a term rule and a finite total

An arithmetic sequence describes quantities that change by the same amount each time. This can model situations such as a regular weekly saving, rows of seats that increase by a fixed number, or a steadily changing measurement. A sequence lists the values one at a time; a series adds a chosen number of those values. In this lesson, the term number is a positive integer, and a finite series includes a specified first and last term. The goal is to choose a model, use it accurately, and explain what its answer means.

What you will learn

Recognize an arithmetic sequence from its constant difference between consecutive terms.
Use the first term and common difference to model a term number and find a specified term.
Use arithmetic series formulas to calculate a finite total and interpret the result in context.
Connect a rule to a term list, a discrete graph, and a contextual model.

1. Prior knowledge: terms and differences

A sequence is an ordered list. Its first term is denoted by

u_1

, its second by

u_2

, and its

n

th term by

u_n

. The subscript identifies a position; it is not a multiplication sign. The term number

n

is a positive integer.

To test whether a sequence is arithmetic, subtract each term from the next. If the difference is always the same, the sequence is arithmetic. That fixed difference is called the common difference, denoted by

d

. It may be positive, negative, or zero.

For example, in the list CAD 14, 10, 6, 2, each term is

4

less than the previous one, so

d=-4

. A negative common difference means the terms decrease as the term number increases. If the differences are not constant, this arithmetic model does not fit the list.

d=u_{n+1}-u_n

Find the difference in the order next term minus previous term.
Check more than one pair of consecutive terms.
The position of a term and its value are different quantities.

2. The term rule and its representations

Starting at

u_1

, reaching

u_n

requires

n-1

equal changes. This gives a direct rule for the value at any positive-integer position. The formula works for increasing, decreasing, and constant arithmetic sequences.

The same model can be written recursively: begin with

u_1

, then add

d

to obtain each next term. A recursive rule helps generate terms in order; the direct rule is more useful when a distant term is required.

A term table displays the values against their positions. On a graph, plot the points

(n,u_n)

for positive-integer

n

. They lie along a straight-line pattern, but only the integer-position points belong to the sequence. The corresponding straight-line expression helps describe the pattern, but non-integer positions are not sequence terms.

In a context, identify what one term represents, what one increase in term number means, and what the common difference represents. For example, if

u_n

is the number of seats in row

n

, then

d

is the change in seats from one row to the next. Check that the chosen range of

n

makes sense in the situation.

u_n=u_1+(n-1)d

Use the direct rule when the term number is known.
Interpret $u_1$ , $d$ , and $n$ in the context and include units where relevant.
A sequence graph consists of discrete points, not every point on a continuous line.

3. Finite arithmetic series: adding terms

An arithmetic series is the sum of a finite number of terms from an arithmetic sequence. For instance, adding the first

n

terms gives

S_n=u_1+u_2+\cdots+u_n

. Pairing the first and last terms, then the second and second-last, shows why the total depends on the number of terms and the two end terms: each pair has the same sum.

The sum can be calculated using the first term and last term, or using the first term, common difference, and number of terms. These are equivalent formulas. The first is convenient when both ends are known; the second is useful when the common difference is known.

In a model, be clear about whether the total includes the first term and how many terms are included. If the sequence begins at term

u_1

and ends at term

u_n

, the series contains

n

terms. A calculator can evaluate a long sum, but writing down the model and confirming the number of terms prevents an incorrect total.

S_n=\frac{n}{2}(u_1+u_n)=\frac{n}{2}\bigl(2u_1+(n-1)d\bigr)

A sequence gives individual values; a series adds a finite set of them.
Use the actual final included term as $u_n$ .
Check that the number of terms matches the context.

4. Technology and a reliable solution check

A graphing calculator or spreadsheet can list terms, plot

(n,u_n)

, and evaluate a finite sum. Use it to check a model, not to replace the reasoning: enter the first term, common difference, and intended term numbers, then compare the displayed values with the formula.

For a graph, set the horizontal variable to term number and inspect integer positions. For a sum, check the first and last entries in the list and count how many entries are included. A calculator's sum command may use different start and end conventions, so verify what it has added.

For an exam-style response, state what each variable represents, identify the constant difference, select the appropriate formula, substitute values, and give the result with units or a contextual statement. A final check can be made by generating nearby terms or by comparing the series total with the average of its first and last terms multiplied by the number of terms.

S_n=n\left(\frac{u_1+u_n}{2}\right)

Use technology to check numerical values and visualize discrete terms.
Show the sequence or series model before reporting calculator output.
Check sign, term count, endpoint inclusion, and units.

A term table for a decreasing arithmetic sequence

Term number $n$	Term value $u_n$
$1$	$14$
$2$	$10$
$3$	$6$
$4$	$2$

Worked example

Finding a distant term

A sequence begins

23, 18, 13, 8,\ldots

. Find its

20

th term.

Identify the pattern
The first term is $23$ . Consecutive terms decrease by $5$ , so the common difference is $-5$ .
$u_1=23,\quad d=-5$
Use the direct rule
There are $20-1$ changes from the first term to the twentieth. Substituting into the arithmetic term rule gives the required value.
$u_{20}=23+(20-1)(-5)$
Evaluate
The result is the value at position $20$ . Its being below the first term is consistent with the negative common difference.
$u_{20}=23-95=-72$

Answer: The twentieth term is

-72

.

Check: The term at position

n

is

28-5n

, so at

n=1

it is

23

and at

n=20

it is

-72

. The rule reproduces both the starting value and the result.

Worked example

Modeling a finite total

A student saves CAD 12 in the first week and increases the amount saved each week by CAD 3. Find the total saved over 10 weeks, assuming this pattern continues.

Define the sequence
Let $u_n$ be the amount saved in week $n$ , measured in CAD. The first amount is CAD 12, the weekly increase is CAD 3, and the number of terms is $10$ .
$u_1=12,\quad d=3,\quad n=10$
Find the last weekly amount
Use the term rule to calculate the amount saved in week $10$ . There are nine increases after week $1$ .
$u_{10}=12+(10-1)(3)=39$
Add the ten amounts
The sum is the number of weeks multiplied by the average of the first and last weekly amounts.
$S_{10}=\frac{10}{2}(12+39)=255$

Answer: The student saves CAD 255 over 10 weeks.

Check: The weekly amounts run from CAD 12 to CAD 39, so their average is CAD 25.50. Ten amounts at this average total CAD 255, matching the series calculation.

Worked example

Finding how many terms are included

A hall has 16 seats in its first row. Each successive row has 4 more seats than the previous row. The first 12 rows are counted. Find the total number of seats in those rows.

Describe the model
The number of seats in a row forms an arithmetic sequence. There are $12$ terms, beginning with $16$ , and each row adds $4$ seats.
$u_1=16,\quad d=4,\quad n=12$
Calculate the final row
From the first row to the twelfth row there are $11$ equal increases. This gives the number of seats in row $12$ .
$u_{12}=16+(12-1)(4)=60$
Calculate the total
Pairing the first and last row values gives an average of $38$ seats per row. Multiply this average by the $12$ rows.
$S_{12}=\frac{12}{2}(16+60)=456$

Answer: The first 12 rows contain 456 seats.

Check: The row counts form

16,20,24,\ldots,60

. The first and last average is

38

, and

12\times 38=456

. This checks both the endpoint and the total.

Common mistakes and how to avoid them

Using

n

changes instead of

n-1

changes when finding the

n

th term.

Correction: The first term is already at position

1

, so reaching position

n

requires

n-1

steps.

Treating a negative common difference as a negative term in every position.

Correction: The sign of

d

describes the change between terms. Calculate the term itself from the first term and position.

Using the number of terms as the final term value in a sum.

Correction: Find the value of the last included term first, then use it as

u_n

in the sum formula.

Including an extra term when entering a sum in technology.

Correction: List or identify the first and last included positions, then confirm that exactly the intended number of terms has been added.

Lesson summary

An arithmetic sequence has a constant difference between consecutive terms.
The term rule uses the first term and $n-1$ equal changes.
A finite arithmetic series adds a specified number of terms.
A discrete graph, a term table, and a contextual description can all represent the same arithmetic model.
Check that the term count, endpoints, sign, and units match the situation.

Check your understanding

Question 1

An arithmetic sequence has first term

7

and common difference

6

. What is its fifth term?

$25$
$31$
$37$
$43$

Show answer and explanation

25

From the first to the fifth term there are four increases:

7+4(6)=31

is not correct arithmetic? Re-evaluate:

7+24=31

. Thus the correct option is

31

.

Question 2

Find the sum of the first 8 terms of

5, 9, 13,\ldots

.

$152$
$136$
$144$
$160$

Show answer and explanation

152

The eighth term is

5+7(4)=33

. The sum is

\frac{8}{2}(5+33)=152

.

Key terms

Sequence: An ordered list of values, with each value attached to a position.
Term: A value in a sequence; $u_n$ denotes the value at position $n$ .
Common difference: The fixed amount added to move from one term to the next in an arithmetic sequence.
Series: The sum of a specified set of terms from a sequence.

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Published by DoAssignment. This AI-assisted lesson follows International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL, study topic SL 1.2. It is a study resource, not an official curriculum publication.

Before publication, the draft is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability, and then requires administrator approval. Errors can still occur, so corrections are welcomed.

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SL 1.2 · Model arithmetic sequences and series

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

From repeated change to a term rule and a finite total

What you will learn

1. Prior knowledge: terms and differences

2. The term rule and its representations

3. Finite arithmetic series: adding terms

4. Technology and a reliable solution check

A term table for a decreasing arithmetic sequence

Finding a distant term

Modeling a finite total

Finding how many terms are included

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through IB AA SL

About this lesson and its review