C. Discrete Functions · Sequences and Series

Sequences as Discrete Functions

C1.1 C1.3About 70 minutes

Learning goals

I can explain what a sequence is and use the notation t₁, t₂, …, tₙ.
I can explain why a sequence is a function whose domain is the natural numbers.
I can write terms of a sequence from its general term or from function notation.
I can tell a discrete function from a continuous function, from a graph or from a situation.

Words to know

sequence
An ordered list of numbers, such as 12, 14, 16, 18, …

term
One number in a sequence. t₁ is the first term, t₂ the second, and tₙ the term in position n.

general term (nth term)
A formula for tₙ in terms of n, such as tₙ = 3n − 1. It lets you find any term directly.

finite / infinite sequence
A finite sequence stops (2, 4, 6, 8). An infinite sequence goes on forever, shown by three dots (2, 4, 6, 8, …).

natural numbers (ℕ)
The counting numbers {1, 2, 3, 4, …}. They are the positions in a sequence.

discrete function
A function whose domain is made of separate values (such as natural numbers). Its graph is a set of separate points.

continuous function
A function whose domain is an interval of real numbers. Its graph is an unbroken line or curve.

Counting in steps

A town is setting up chairs for an outdoor concert at its winter carnival. Row 1 has 12 chairs. Each row behind it has 2 more chairs than the row in front: 12, 14, 16, 18, and so on. How many chairs are in row 20? How many chairs are there altogether?

A list of numbers in a definite order, like 12, 14, 16, 18, …, is called a sequence. This unit is all about sequences: how to describe them, how to find any term without listing them all, and how to add them up. By the end you will be able to answer both questions about the chairs in about a minute.

Remember from Grades 9 and 10

In Grade 9 you met the natural numbers ℕ = {1, 2, 3, …}, used first differences to tell linear from non-linear relations, and saw sequences such as 1/2, 3/4, 7/8, … that get closer and closer to a limit. In Grade 10 you used second differences to spot quadratic relations. Earlier in this course, in the unit Functions and Equivalent Expressions, you used function notation f(x) and described domain and range. Today you will put these ideas together.

Terms and notation

Each number in a sequence is a term. We name the terms with the letter t and a subscript that gives the position:

t₁ = 12 is the first term,
t₂ = 14 is the second term,
tₙ is the term in position n. It is called the general term or nth term.

A sequence can be finite (it stops, like 12, 14, 16, 18, 20) or infinite (it never stops; we show this with three dots, like 12, 14, 16, …).

If you know a formula for tₙ, you can find any term without listing the ones before it. For the chairs, the formula is tₙ = 2n + 10. Check: t₁ = 2(1) + 10 = 12 and t₂ = 2(2) + 10 = 14. Row 20 has t₂₀ = 2(20) + 10 = 50 chairs.

A sequence is a function

A sequence pairs each position n with exactly one term tₙ:

(1, 12), (2, 14), (3, 16), (4, 18), …

Every position gets exactly one term, so a sequence is a function. Its domain is the set of natural numbers {1, 2, 3, …} (or {1, 2, …, N} for a finite sequence), and its range is the set of terms. That means we can also use function notation: write f(n) = 2n + 10 instead of tₙ = 2n + 10. Then f(20) = 50 means "the 20th term is 50." The two notations say exactly the same thing: f(n) = tₙ.

A mapping diagram for tₙ = 3n − 1. Each position (left) is sent to exactly one term (right), which is what makes a sequence a function.

n (position)	1	2	3	4	5
tₙ = 2n + 10 (chairs in row n)	12	14	16	18	20

The concert chairs as a table of values. The top row is the domain, the bottom row is the range.

Discrete or continuous?

Look at f(x) = 2x in two different settings.

If the domain is the natural numbers, the function only has values at 1, 2, 3, …. Its graph is a set of separate, equally spaced points. We call it a discrete function. Do not join the dots: there is no 2.5th term.
If the domain is the real numbers, every x has a value, including 2.5 and √2. The graph is an unbroken straight line. We call it a continuous function.

Every sequence is a discrete function. In real situations, ask: can the input take any value in an interval, or only separate values? Counting things (people, chairs, rows, days) is usually discrete. Measuring things that change smoothly (time, distance, temperature) is usually continuous.

The same rule, f(x) = 2x, with two different domains. On the natural numbers it is discrete (left: dots only). On the real numbers it is continuous (right: a solid line).

Worked example

Example 1: Writing terms from the general term

A sequence has general term tₙ = 3n − 1.

(a) Write the first five terms. (b) Find t₁₂.

Substitute n = 1, 2, 3, 4, 5 into 3n − 1.
t₁ = 3(1) − 1 = 2, t₂ = 3(2) − 1 = 5, t₃ = 8, t₄ = 11, t₅ = 14.
(b) t₁₂ = 3(12) − 1 = 36 − 1 = 35.

(a) 2, 5, 8, 11, 14 (b) t₁₂ = 35

Worked example

Example 2: Function notation

A sequence is defined by f(n) = n² − 2n, n ∈ ℕ. Write the first five terms and find f(10).

f(1) = 1 − 2 = −1
f(2) = 4 − 4 = 0
f(3) = 9 − 6 = 3, f(4) = 16 − 8 = 8, f(5) = 25 − 10 = 15
f(10) = 100 − 20 = 80.
Notice "n ∈ ℕ" (read "n is an element of the natural numbers"). It tells you the function is discrete.

−1, 0, 3, 8, 15, and f(10) = 80.

Worked example

Example 3: Is a number a term?

For tₙ = 3n − 1, is 101 a term of the sequence? Is 150?

Set the general term equal to the number and solve for n.
3n − 1 = 101 → 3n = 102 → n = 34. This is a natural number, so 101 is a term: it is t₃₄.
3n − 1 = 150 → 3n = 151 → n = 50.33… This is not a natural number, so 150 is not a term.

101 = t₃₄ is a term; 150 is not a term.

Worked example

Example 4: Discrete or continuous?

Decide whether each relation is discrete or continuous, and explain.

(a) The cost C(n) of n concert tickets at $15 each. (b) The depth of snow on the ground during a 6-hour snowstorm. (c) The noon temperature recorded each day in January.

(a) You can only buy whole tickets: 0, 1, 2, 3, …. The domain is separate values, so it is discrete. Graph: separate points.
(b) Time can be any value from 0 h to 6 h, and the depth changes smoothly. It is continuous. Graph: a smooth curve.
(c) The temperature itself changes continuously, but this data set has one value per day (days 1 to 31). The data are discrete: plot 31 points.

(a) discrete, (b) continuous, (c) discrete data.

Just for fun

Why did the sequence and the continuous function stop hanging out?

The sequence said it needed some space between them.

Hands-on activity

Toothpick squares

You need: about 30 toothpicks (or matches with the heads cut off by a grown-up, or cotton swabs), paper and a pencil.

Make one square with 4 toothpicks. Then build a row of 2 squares that share a side, then a row of 3, then 4, then 5.
Make a table: n (number of squares) and tₙ (number of toothpicks).
Find a formula for tₙ. Hint: the first square needs 4; each new square adds the same number.
Use your formula to predict how many toothpicks a row of 25 squares needs. Is 100 a term of your sequence?
Plot your five points on grid paper. Should you join them with a line? Why or why not?

What to look for: the terms are 4, 7, 10, 13, 16 and tₙ = 3n + 1. A row of 25 squares needs 3(25) + 1 = 76 toothpicks. 3n + 1 = 100 gives n = 33, so 100 is a term (t₃₃). The points lie on a line but should not be joined: you cannot build 2.5 squares, so the function is discrete.

Your turn

Write the first four terms of tₙ = 5n + 2.

For tₙ = 5n + 2, find t₂₀.

Write the first five terms of f(n) = 2n² − 1, n ∈ ℕ.

For tₙ = 100 − 4n, find t₁ and t₂₅.

Is 86 a term of tₙ = 4n + 6? If so, which one?

Is 50 a term of tₙ = 4n + 6?

Discrete or continuous: (a) the number of maple trees tapped each day; (b) the volume of sap in a bucket during the afternoon?

More practice: warm-up

In the sequence 3, 9, 27, 81, …, what is t₃?

Is 1, 4, 9, 16, 25 a finite or an infinite sequence?

For f(n) = 7n, find f(6).

What is the domain of an infinite sequence?

Should you join the points on the graph of a sequence? Why?

More practice: on your own

Write the first five terms of tₙ = n(n + 1)/2.

A sequence has f(n) = (−1)ⁿ × n. Write the first six terms.

Is 200 a term of tₙ = 6n − 4? Explain.

Is 75 a term of tₙ = n² + 3? Explain.

A hockey arena sells popcorn for $4 a bag. Write a function for the cost of n bags, state its domain, and say whether it is discrete or continuous.

Challenge

The sequence tₙ = 2n² − 3n has a term equal to 104. Which term is it? Is 90 a term?

Give an example of a situation where a continuous model is used even though the real data are discrete, and explain why that is reasonable.

Find a general term tₙ for 1/2, 2/3, 3/4, 4/5, …, and find t₉₉.

Check your understanding

Quiz

For tₙ = 4n − 3, what is t₁₀?

What is the domain of the sequence 5, 10, 15, 20, …?

Which graph shows a discrete function?

f(n) = n² + n. What is f(4)?

Which situation is best modelled by a continuous function?

Is 58 a term of tₙ = 3n + 4?

For parents and teachers

Big idea: Expectation C1.1 asks students to see sequences as discrete functions and to tell discrete from continuous functions (the curriculum's own example is f(x) = 2x on the natural numbers versus the real numbers). C1.3 connects the nth-term formula tₙ to function notation f(n).

How to help: Point out sequences in daily life: seat numbers in rows, a weekly allowance that grows, bus departure times. Ask "What would the 10th one be? The 100th?" and "Would it make sense to join these points with a line?"

Common difficulties:

Mixing up the position n and the term tₙ. In t₅ = 23, 5 is the position and 23 is the value.
Joining the points of a sequence's graph. The dots should stay separate.
Accepting a non-whole n when checking whether a number is a term. If n is not a natural number, the number is not a term.
Skipping the check. Encourage your teen to substitute every answer back into the formula; it takes ten seconds and catches most slips.