9.1: Introduction to Sequences Grade Distribution 1st 3rd A 2 4 B 10 C D 1 F 3 100+ No Shows Range 47-97 51-95 Avg 72.79 80.91 2/21/2019 8:32 PM 9.1: Introduction to Sequences

Introduction to Sequences Section 9.1 Precalculus PreAP/Dual Revised ©2017 Viet.dang@humbleisd.net 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences The Factorial Factorial is an integer greater than or equal to zero in which uses the equation, 𝒏 𝒏−𝟏 ! Represented by the symbol, “!” 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Definitions Sequence is an ordered list of numbers. (Known as the Fibonacci sequence) Finite sequence has a limited number of terms, such as {1, 2, 3, 4}. Infinite which continues without end such as {1, 2, 3, 4, …} Series is where a sequence is added. A series can be finite or infinite 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 1 Evaluate 𝟗! 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 2 Evaluate 𝟑!𝟕! 𝟒! 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Your Turn Evaluate 𝟏𝟐! 𝟏𝟎! 2/21/2019 8:32 PM 9.1: Introduction to Sequences

Understanding Sequences This is a sequence having the rule, 𝒂𝒏=𝟑𝒏, The domain gives the relative position of each term. 𝒏 𝒂𝒏 𝟏 𝟐 𝟑 𝟒 𝟓 DOMAIN: The range gives the terms of the sequence. 𝟑 𝟔 𝟗 𝟏𝟐 𝟏𝟓 RANGE: 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 3 Write the first six terms of the sequence of 𝒂 𝒏 =𝟐𝒏+𝟑. 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 4 Write the first six terms of the sequence of 𝒂 𝒏 = −𝟏 𝒏+𝟏 𝟐 𝒏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 5 Write the first five terms of the sequence of 𝒂 𝒏 = 𝒆 𝒏 𝒏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Your Turn Write the first six terms of the sequence of 𝒂 𝒏 = 𝒏−𝟏 𝒏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Definition Factorial is a symbol used to multiply the previous highest number to the lowest number consecutively. Recursive formula is a rule where the sequence is assigned a value to the first few terms and specified the nth term 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 6 A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =𝟑, 𝒂 𝒏 =𝟔+ 𝒂 𝒏−𝟏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 7 A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =𝟓, 𝒂 𝒏 =𝟒 𝒂 𝒏−𝟏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Your Turn A sequence is defined recursively. Determine the first five terms of 𝒂 𝟏 =−𝟔, 𝒂 𝒏 =𝒏+ 𝒂 𝒏−𝟏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 8 Determine the equation of this sequence: 𝟏, 𝟏 𝟓 , 𝟏 𝟐𝟓 , 𝟏 𝟏𝟐𝟓 ,… 𝒏 term Term 𝟏 𝟏 𝟏 𝟐 𝟏 𝟓 𝟑 𝟏 𝟐𝟓 𝟒 𝟏 𝟏𝟐𝟓 𝒏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 9 Determine the equation of this sequence: 𝟏𝟎, −𝟐𝟎,𝟑𝟎,−𝟒𝟎,… 𝒏 term Term 𝟏 𝟏𝟎 𝟐 −𝟐𝟎 𝟑 𝟑𝟎 𝟒 −𝟒𝟎 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 10 Determine the equation of this sequence: 𝟏 𝟐𝟎 , 𝟐 𝟐𝟏 , 𝟑 𝟐𝟐 , 𝟒 𝟐𝟑 ,… 𝒏 term Term 𝟏 𝟏 𝟐𝟎 𝟐 𝟐 𝟐𝟏 𝟑 𝟑 𝟐𝟐 𝟒 𝟒 𝟐𝟑 𝒏 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Sigma Notation upper limit of summation Is read as “the sum from k equals 1 to 5 of 3k.” 5 k = 1 ∑ 3k 3 + 6 + 9 + 12 + 15 = ∑ 3k 5 k = 1 index of summation lower limit of summation 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 11 Write out the term, 𝒌=𝟏 𝒏 𝒌! 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Example 12 Express the sum of 𝟏 𝟐 + 𝟐 𝟐 + 𝟑 𝟐 + …𝟗 𝟐 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Your Turn Write out the term, 𝒌=𝟏 𝟏𝟎 𝟏 𝒌 2/21/2019 8:32 PM 9.1: Introduction to Sequences

9.1: Introduction to Sequences Assignment Page 613 7-23 EOO, 37-55 odd, 59-65 odd, 79-87 odd 2/21/2019 8:32 PM 9.1: Introduction to Sequences