3: Number Patterns

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Course Goals and Anticipated Outcomes for This Chapter:

By the end of this Chapter, students should be able to:

Identify arithmetic, geometric, quadratic, and recursively defined sequences.
Derive and use explicit formulas and recurrence relations.
Evaluate finite arithmetic and geometric sums using algebraic and visual methods.
Construct valid proofs using ordinary and strong induction.
Use finite-difference tables to find polynomial models and explain their limitations.

THINKING OUT LOUD

Consider the following sequence of numbers in which only the first two terms are given: $1,3,\cdots, . $ Create four different number patterns with the first two terms $1, 3 $ by writing out the next four terms. In each case, explain the rule for your pattern. What happens if the first four terms are given as $1, 3, 5, 7, \cdots$? How many possibilities are there?

THINKING OUT LOUD

What is the perimeter of the design by joining $n$ regular hexagons in a row? How can you prove your prediction?

$row of hexagons.png$

Numbers can be organized into many different sequences. Most of these sequences have patterns we can use to predict the next number. Misunderstandings can occur when we list only a few numbers in a sequence. For example, $3,5,7..$, the next term could be either $9$ (sequence of odd integers) or $11$ (sequence of prime numbers). Therefore, it is wise to define sequences using an explicit formula for the $n^{th}$ term.

There are many types of patterns, but we will be looking at the following:

Arithmetic sequences
Finite sums of arithmetic sequences
Geometric sequences
Finite sums of geometric sequences
Some other types of sequences

All sequences, regardless of how they progress, have terms. To denote which term we wish to consider, we use $n$. So, if we say that $n = 3$, we are considering the third term in a sequence. Normally, the first term of a sequence is denoted by $a$. If we set $a = 23$, the first term of the sequence is 23.

So, without further ado, let's be off!

Definition: Sequence

A sequence is a function whose domain consists of consecutive integers, often the positive integers.

A sequence may be written as

\[ (a_n)_{n\geq1} \]

where $a_n$ denotes its $n$th term.

Its first term is $a_1$, and its second term is $a_2$.

Example $\PageIndex{1}$

For $n$ hexagons, each with side length $s$, arranged in a chain with exactly one shared side between consecutive hexagons and no other shared sides,

$ P_n=6ns-2(n-1)s. $

Thus,

$ \boxed{P_n=(4n+2)s.} $

Notations

Terms: the numbers in a sequence

When considering a specific term: $n = x$, where x is a whole number.
The first term in a sequence: $a$

Thumbnail: Derivation of triangular numbers from a left-justified Pascal's triangle. 9Cc BY-SA 4.0; Cmglee).

Thanks to Thomas Thangarajah for sharing his hexagonal drawing.

Contributor

Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)

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Definition: Sequence

Example \(\PageIndex{1}\)

Notations

Contributor