Explicit & Recursive Formulas

 A Sequence is a list of things (usually numbers) that are in order.  2 Types of formulas:  Explicit & Recursive Formulas

 A list of numbers that related to each other by a rule.  The terms are the numbers that form the sequence.  Goes from one term to the next by adding and subtracting.  Two types of sequences with two types of formulas. 1. Arithmetic: Explicit & Recursive 2. Geometric: Explicit & Recursive

 A sequence in which the common difference between terms is always the same number.  ADD to get to the next term.  2 types of formula to find either the NEXT number or A NUMBER in the sequence. Explicit vs Recursive

 Finds the NEXT Term in the sequence a n = a n-1 + d Previous Term Common Difference

 Find the common difference. 1) 3, 5, 7, 9, 11 2) -2, -4, -6, -8, -10 3) -3, 0, 3, 6, 9 4) 19, 10, 1, -8, -17

NEXT Number = CURRENT Term + d a n = a n-1 + d

1) Find the 4 th term given: a 4 = ?, a 3 = 6, d = 2 2) Find the 3 rd term given: a 3 = ?, a 2 = 17, d = -7 3) Find the 6 th term given: a 6 = ?, a 5 = 2, d = -5

4) 12, 9, 6, 3, 0, … 5) 56, 61, 66, 71, … 6) -14, -24, -34, -44, …

First Term Common Difference One less term than the term number

 Consider the sequence {6, 17, 28, 39, 50, …}  Find the 12 th term.  Find the 50 th term.  Find the 100 th term.

 A bag of dog food weighs 8 pounds at the beginning of day 1. Each day, the dogs are feed 0.1 pound of food. How much does the bag of dog food weigh at the beginning of day 30?

1. Determine if the sequence is Arithmetic -Do you add or subtract by the same amount from one term to the next? 2. Ask if you want to find the next term or a term in the sequence? 3. Find the common difference. -The number you add or subtract 4. Create a Recursive or Explicit formula -State first term or previous term,

 A sequence in which the ratio between consecutive terms is always the same number.  A geometric sequence is formed by multiplying a term in the sequence by a fixed number to find the next term.  Common Ratio- the fixed number that you multiply each term in the sequence.

 A sequence in which the common ratio between terms is always the same number.  MULTIPLY to get to the next term.  2 types of formula to find either the NEXT number or A NUMBER in the sequence. Explicit vs Recursive

 Finds the NEXT Term in the sequence a n = a n-1 ∙ r Previous Term Common Ratio

1. 750, 150, 30, 6,… , -36, 144, -576,… 3. 8, -24, 72, -216,…

1. 750, 150, 30, 6,… 2. -3, -6, -12, -24,… 3. 9, -36, 144, -576,…

First Term Common Ratio One less term than the term number

1. 750, 150, 30, 6,… 2. -3, -6, -12, -24,… 3. 9, -36, 144, -576,… 4. 8, -24, 72, -216,…

1. Determine if the sequence is Geometric -Do you multiply or divide by the same amount from one term to the next? 2. Ask if you want to find the next term or a term in the sequence? 3. Find the common ratio. -The number you multiply or divide 4. Create a Recursive or Explicit formula -State first term or previous term,

 Arithmetic Graphs: LINEAR  Geometric Graphs: EXPONENTIAL

 A indicated Sum of terms of a sequence is called a Series  OR….  A Sum of an infinite sequence it is called a "Series"  (it sounds like another name for sequence, but it is actually a sum).

Arithmetic SequenceArithmetic Series 3, 6, 9, 12, , -1, (-1) + 2

Sum of Numbers in Sequence Must Know first term, last term, and number of terms Number of terms First term nth term