3-5 Arithmetic Sequences as Linear Functions

3-5 Arithmetic Sequences as Linear Functions A sequence is an ordered list of numbers.   The sum of the terms of a sequence is called a series. While some sequences are simply random alues, other sequences have a definite pattern that is used to arrive at the sequence's terms. Two such sequences are the arithmetic and geometric sequences.  Let's investigate the arithmetic sequence.

What is an arithmetic sequence? EQ: What can graphs tell us? I will be able to: Recognize arithmetic sequences Relate arithmetic sequences to linear functions Vocabulary: Sequence Terms of the sequence Arithmetic sequence Common difference

What is an arithmetic sequence? If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an  arithmetic sequence.   The number added to each term is constant (always the same). The fixed amount is called the common difference, d,referring to the fact that the difference between two successive terms yields the constant value that was added.   To find the common difference, subtract the first term from the second term. Notice the linear nature of the scatter plot of the terms of an arithmetic sequence.  The domain consists of the counting numbers 1, 2, 3, 4, ... and the range consists of the terms of the sequence.  While the x value increases by a constant value of one, the y value increases by a constant value of 3 (for this graph).      

Some examples…. Determine whether –15, –13, –11, –9, ... is an arithmetic sequence. Explain. . Determine whether is an arithmetic sequence. Explain. Determine whether 2, 4, 8, 10, 12, … is an arithmetic sequence.

More examples…. Find the next three terms of the arithmetic sequence 58, 63, 68, 73, …. Find the next three terms of the arithmetic sequence –8, –11, –14, –17, …. Find the next 3 terms.

More applications – Find the nth term . Write an equation for the nth term of the arithmetic sequence 1, 10, 19, 28, … . an = a1 + (n – 1)d Formula for the nth term a1 = 1, d = 9 Find the 12th term in the sequence.

The graph of an arithmetic sequence is linear. Graph the first five terms of the sequence. Which term of the sequence is 172? an = 9n – 8

Real World Examples MONEY The arithmetic sequence 2, 7, 12, 17, 22, … represents the total number of pencils Claire has in her collection after she goes to her school store each week. A. Write an equation for the nth term of the sequence. Which graph shows the first five terms of the sequence? What is the 97th term of the sequence? B. Find the 12th term in the sequence. A. B.

Another Real World Example NEWSPAPERS The arithmetic sequence 12, 23, 34, 45, ... represents the total number of ounces that a bag weighs after each additional newspaper is added. A. Write a function to represent this sequence. an = a1 + (n – 1)d a1 = 12 and d = 11 Graph the function an = 11n + 1 and determine the domain. Write a function to represent this sequence.

What is an arithmetic sequence? SHIPPING The arithmetic sequence 22, 40, 58, 76, … represents the total number of ounces that a box weight after each additional bottle of salad dressing is added. A. Write a function to represent this sequence. B. Graph the function an = 18n + 4 and determine the domain of the sequence.