ARITHMETIC SEQUENCES

SEQUENCE  What is a sequence?  “A list of things (usually numbers) that are in order.”  We are free to define what order that is ! They could go forwards, backwards... or they could alternate... or any type of order we want!  Interchangeable: Term, Element, and Member

FINITE OR INFINITE SEQUENCE  Sequence can be BOTH Finite or Infinite. EXAMPLES Below  {1, 2, 3, 4,...} is a very simple sequence (and it is an infinite sequence )  {20, 25, 30, 35,...} is also an infinite sequence  {1, 3, 5, 7} is the sequence of the first 4 odd numbers (and is a finite sequence )  {1, 2, 4, 8, 16, 32,...} is an infinite sequence where every term doubles  {a, b, c, d, e} is the sequence of the first 5 letters alphabetically  {f, r, e, d} is the sequence of letters in the name "fred"  {0, 1, 0, 1, 0, 1,...} is the sequence of alternating 0s and 1s (yes they are in order, it is an alternating order in this case)

o There are many different types of sequences depending on the relationship of the terms of the sequence. o Geometric Sequence o Fibonacci Sequence o Triangular Number Sequence

Below are examples of Arithmetic Sequences. What is the pattern? Can you determine the next three terms? EX 1 -5, 2, 9, 16,... EX 2 12, 10.5, 9, 7.5, 6,... 23, 30, , 3, 1.5 What is being added to each term to get the next term? How would you define an Arithmetic Sequence?

In arithmetic sequence, the difference between one term and the next is a constant. An arithmetic sequence can be built by adding each term to that constant. Difference between the 1 st term and 2 nd term is the same as the 99 th term and 100 th term.

By analyzing the collection of objects, what would the next figure look like? can you create an Arithmetic Sequence ? EX 3

3, 5, 7, 9,... Now let’s look at things you need to know when working with arithmetic sequences: d refers to the common difference between terms A (1 ) the value of the first term in the sequence A(n ) the value of the nth term in the sequence

Now let’s look at function notation that can be used to identify the terms in a sequence. If the sequence is 8, 14, 20, 26,... A(1) = A(2) = A(3) = A(4) = A(2) = A(1)+6 A(3) = A(2)+6 A(4) = A(3)+6 For this sequence, the function below can be used to find the next term A ( n ) = A ( n -1)+6 given A (1)=8

The function that can be used to find the next term of the sequence is called the RECURSIVE FORM of the Sequence. The recursive form of an arithmetic sequence is A ( n ) = A ( n -1)+ d given a term and its position in the sequence What is the recursive form of these arithmetic sequences? EX 1 -5, 2, 9, 16,... EX 2 12, 10.5, 9, 7.5, 6,... A ( n ) = A ( n -1)+7 ; A (1) = – 5 A ( n ) = A ( n -1) – 1.5 ; A (1) = 12

Recursive form is helpful in creating a sequence but what if you want to find the 50 th term of a sequence and you only know the first 5 terms. We need another form. Let’s look at how the sequence 3, 7, 11, 15, 19,... was created.

3, 7, 11, 15, 19,... Term Term Value A(1)A(1)A(1)A(1) 3 = 3 3 = 3 A(1) + 4 (0) A(1) + 4 (0) A(1) + 4 (1 – 1) A(1) + 4 (1 – 1) A(2)A(2)A(2)A(2) = = 7 A(1) + 4 (1) A(1) + 4 (1) A(1) + 4 (2 – 1) A(1) + 4 (2 – 1) A(3)A(3)A(3)A(3) = = 11 A(1) + 4 (2) A(1) + 4 (2) A(1) + 4 (3 – 1) A(1) + 4 (3 – 1) A(4)A(4)A(4)A(4) = = 15 A(1) + 4 (3) A(1) + 4 (3) A(1) + 4 (4 – 1) A(1) + 4 (4 – 1) A(5)A(5)A(5)A(5) = = 19 A(1) + 4 (4) A(1) + 4 (4) A(1) + 4 (5 – 1) A(1) + 4 (5 – 1) A(50) How would you find A(50)? A(n) How would you find A(n)?

The function that can be used to find the n term of a sequence is called the EXPLICIT FORM of the Sequence. For an arithmetic sequence, the explicit form is A ( n ) = A (1) + d ( n -1) What is the explicit form of these arithmetic sequences? EX 1 -5, 2, 9, 16,... EX 2 12, 10.5, 9, 7.5, 6,... A ( n ) = – 5 + 7( n -1) A ( n ) = 12 – 1.5 ( n -1) th