Coordinate Algebra Arithmetic and Geometric Sequences Learning Target: Students can use the explicit formula to find the n th term of a sequence.

Arithmetic Sequences (Common difference) Geometric Sequences (common ratio) Explicit Formula Recursive Formula Explicit Formula Recursive Formula

Each distance is 0.2 miles greater than the previous distance. Example: The distance between you and a lightning strike can be approximated by using a sequence. Times (s)Distance (mi) *When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. A sequence is a list of numbers that may form a pattern. Each number in a sequence is called a term.

Arithmetic Sequence A sequence of terms that have a common difference (d) between them -Subtract terms right to left

12, 8, 4, 0, … Step 1: Find the difference between successive terms. Step 2: Use the common difference to find the next 3 terms. Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence. Add -4 to each term to find the next term. The sequence is arithmetic and the common difference is = -4, -4 – 4 = -8, -8 – 4 = -12

1, 4, 9, 16, … Step 1: Find the difference between successive terms. The difference between successive terms is not the same, so the sequence is not arithmetic = 4, = 9, = 16

Explicit Formula for Arithmetic Sequence (Finding the nth term)

Explicit Formula for Arithmetic Sequences a 1 + (n-1)d = a n (Allows us to find ANY term in a sequence) a 1 1 st term a 1 = 1 st term nthe number of the term you are looking for n= the number of the term you are looking for dcommon difference d= common difference a n the value of the term that you are looking for a n = the value of the term that you are looking for

Geometric Sequence A sequence of terms that have a common ratio (r) between them -Divide terms right to left

Explicit Formula Formula used to find the n th term of a sequence

Explicit Formula for Geometric Sequence

Arithmetic or Geometric? Example : -22, -15, -8, -1, … Arithmetic d = 7

Arithmetic or Geometric? Example : 7, 4, 1, -2, -5 Arithmetic d = -3

Arithmetic or Geometric? Example : 256, 64, 16, 4, … Geometric r = 1/4

Arithmetic or Geometric? Example : Geometric r = 2/3

Find the common difference, the explicit formula, and the tenth term. 3, 9, 15, 21, … d = 6 a 10 = 57

Find the common ratio, the explicit formula, and the seventh term. 3, 1.5, 0.75, 0.375, …

The fifth term is 1,792. The constant ratio is 4. Write the explicit formula.

Homework Sequence Practice WS