Course Goals and Anticipated Outcomes for This Chapter:
Develop the student:
- ability to understand number patterns and predict the pattern,
- familiarity and facility with a wide range of number patterns and the connection to K-9 curriculum, and
- reasoning using induction and also using finite difference Calculus.
THINKING OUT LOUD
Consider the following sequence of numbers in which only the first two terms are given: 1,3,⋯,⋯. Create four different number patterns having first two terms as 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?
Numbers can be organized into many different sequences. Most of these sequences have patterns which can be used to predict the next number in the pattern. Misunderstandings may occur when we list a few numbers in the 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 in terms of 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
- 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. The first term in a sequence is given by \(a\). So, if we say that \(a = 23\), the first term in the given sequence is 23.
So, without further ado, let's be off!
New Notation & Definitions
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. Image used with permission 9Cc BY-SA 4.0; Cmglee).
Thanks to Thomas Thangarajah for sharing his hexagonal drawing.
Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)