# 6.17 - Sequences

6.17    The student will identify and extend geometric and arithmetic sequences.

Bloom's Level:  Understand Apply

### BIG IDEAS

• I can crack a secret code, figure compound interest, and determine a retirement salary if given the same raise each year.
• I will recognize that patterns exhibit relationships that can be extended, described, and generalized.

### UNDERSTANDING THE STANDARD

• Numerical patterns may include linear and exponential growth, perfect squares, triangular and other polygonal numbers, or Fibonacci numbers.
• Arithmetic and geometric sequences are types of numerical patterns.
• In the numerical pattern of an arithmetic sequence, students must determine the difference, called the common difference, between each succeeding number in order to determine what is added to each previous number to obtain the next number. Sample numerical patterns are 6, 9, 12, 15, 18, …; and 5, 7, 9, 11, 13, …
• In geometric number patterns, students must determine what each number is multiplied by to obtain the next number in the geometric sequence. This multiplier is called the common ratio. Sample geometric number patterns include 2, 4, 8, 16, 32, …; 1, 5, 25, 125, 625, …; and 80, 20, 5, 1.25, …
• Strategies to recognize and describe the differences between terms in numerical patterns include, but are not limited to, examining the change between consecutive terms, and finding common factors. An example is the pattern 1, 2, 4, 7, 11, 16, …

### ESSENTIALS

• What is the difference between an arithmetic and a geometric sequence?
While both are numerical patterns, arithmetic sequences are additive and geometric sequences are multiplicative.

The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to

6.171  Investigate and apply strategies to recognize and describe the change between terms in  arithmetic patterns.

6.172  Investigate and apply strategies to recognize and describe geometric patterns.

6.173  Describe verbally and in writing the relationship between consecutive terms in an arithmetic or geometric sequence.

6.174  Extend and apply arithmetic and geometric sequences to similar situations.

6.175  Extend arithmetic and geometric sequences in a table by using a given rule or mathematical relationship.

6.176  Compare and contrast arithmetic and geometric sequences.

6.177  Identify the common difference for a given arithmetic sequence.

6.178  Identify the common ratio for a given geometric sequence.

### KEY VOCABULARY

geometric sequence, arithmetic sequence, numerical pattern, linear growth, exponential growth, consecutive terms

Updated: Jun 06, 2017