Arithmetic Sequences and Series Section 9.2 Precalculus PreAP/Dual, Revised ©2016 viet.dang@humble.k12.tx.us 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Definitions Arithmetic Sequence is a sequence whose consecutive terms have a common difference 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Is it arithmetic? YES NO −𝟒𝟓, −𝟑𝟎, −𝟏𝟓, 𝟎, 𝟏𝟓… 𝟏, 𝟑, 𝟔, 𝟏𝟐, 𝟏𝟓…. 𝟗 𝟒 , 𝟐, 𝟕 𝟒 , 𝟑 𝟐 , 𝟓 𝟒 … 𝟏 𝟑 , 𝟐 𝟑 , 𝟒 𝟑 , 𝟖 𝟑 , 𝟏𝟔 𝟑 … 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Steps Plug the equation for the nth term of an arithmetic sequence: 𝒂 𝒏 = 𝒂 𝟏 +𝒅 𝒏−𝟏 𝒂 𝟏 is the first sequence number 𝒅 is the common difference To determine the common difference, ensure that all of the values have a common difference Simplify the equation 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 1 Determine the arithmetic sequence equation of 𝟕, 𝟏𝟏, 𝟏𝟓, 𝟏𝟗… 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 1 Determine the arithmetic sequence equation of 𝟕, 𝟏𝟏, 𝟏𝟓, 𝟏𝟗… 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 2 The arithmetic sequence, 𝒂 𝒏 =𝟓−𝟑𝒏 is given. What is the common difference and the first 4 terms? 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn Determine the arithmetic sequence equation of 𝟏𝟗, 𝟏𝟐, 𝟓,−𝟐, −𝟗, … 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 3 Determine the equation whose 𝒏th term of the arithmetic where the first term is 𝟐 and common difference is 𝟑. 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 4 Determine the equation whose 𝒏th term of the arithmetic where the first term is 𝟏𝟕 and common difference is 𝟏𝟕 . 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn Determine the equation whose 𝒏th term of the arithmetic where the first term is −𝟏𝟎 and common difference is −𝟓𝟎. 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 5 Determine its first five terms from the given equation, 𝒂 𝒏 =𝟏𝟐+𝟕𝒏. Then, determine the 20th term. 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 6 If the common difference is 𝟒 and the fifth term is 𝟏𝟓, what is the 10th term of an arithmetic sequence? Do we know what the first term is? 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 6 If the common difference is 𝟒 and the fifth term is 𝟏𝟓, what is the 10th term of an arithmetic sequence? 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn If the common difference is 𝟗 and the 𝟓𝒕𝒉 term is 𝟕𝟑, what is the 10th term of an arithmetic sequence? 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

upper limit of summation 9.2: Arithmetic Sequences Sigma Notation upper limit of summation Is read as “the sum from 𝒏 equals 𝟏 to 𝟓 of 𝟑𝒏.” 𝟓 𝒌=𝟏 ∑ 3𝑛 3 + 6 + 9 + 12 + 15 = ∑ 3n 5 k = 1 index of summation lower limit of summation 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

Steps for Arithmetic Series Identify the number of terms, lower and upper limit of the summation Then, identify the first and last term by plugging the given equation: 𝒌=𝟏 𝒏 𝒂 𝒌 =𝒏 𝒂 𝟏 + 𝒂 𝒏 𝟐 𝒂 𝒌 is the given equation 𝒂 𝟏 is the first term of the sequence 𝒂 𝒏 is the last term of the sequence 𝒏 = term 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 7 Find the following sum, 𝟏+𝟑+𝟓+𝟕+𝟗+𝟏𝟏+𝟏𝟑+𝟏𝟓+𝟏𝟕+𝟏𝟗+𝟐𝟏 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 8 Find the sum of the first 100 terms of the arithmetic sequence, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, … 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn Find the following sum of the total of 14 terms, 𝟏+𝟓+𝟗+…+𝟒𝟗+𝟓𝟑 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 9 Evaluate 𝒌=𝟏 𝟓 𝟑𝒌 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 9 Evaluate 𝒌=𝟏 𝟓 𝟑𝒌 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 10 Evaluate 𝒌=𝟏 𝟏𝟐𝟎 𝟓− 𝟏 𝟒 𝒌 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn Evaluate 𝒌=𝟐 𝟔 𝟐𝒌+𝟏 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Example 11 Determine the amount of terms in this arithmetic series, 𝒂 𝟏 =𝟏𝟗, 𝒂 𝒏 =𝟗𝟔, 𝑺 𝒏 𝑺𝒖𝒎 =𝟔𝟗𝟎 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Your Turn Determine the amount of terms in this arithmetic series, 𝒂 𝟏 =𝟕, 𝒂 𝒏 = 𝟐𝟓, 𝑺𝒏 𝑺𝒖𝒎 =𝟏𝟎 11/26/2018 10:31 PM 9.2: Arithmetic Sequences

9.2: Arithmetic Sequences Assignment Page 622 5-17 EOO, 21-61 odd 11/26/2018 10:31 PM 9.2: Arithmetic Sequences