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Accel Precalc Unit 4: Sequences & Series Lesson #2: Arithmetic Sequences and Partial Sums EQ: What is the formula to find the sum of an arithmetic sequence? Recall: Find the nth term of an Arithmetic Sequence: Arithmetic Sequence --- ordered list of numbers with a common difference called d • What is the Explicit Formula for an Arithmetic Sequence? Can find ANY an a d n 1 1 = _________________ dn a1 d an = _________________ term in sequence *Open Form *Closed Form Ex 1. Write the explicit formula for the sequence 4, 7, 10, 13, … *Open Form an = 4+3n – 3 an = 3n + 1 Then find a13. *Closed Form a13 = 4+3(13 – 1)= 40 a13 = 3(13) +1= 40 What is the Recursive Formula for an Arithmetic Sequence? Can only find the NEXT term in a d , a # n 1 1 an = _________________ sequence Ex 2. The eighth term Common of an arithmetic difference sequence is 75, and theadded twentieth term is 39. to PREVIOUS term Find the first term and the common difference. Give the recursive formula for this sequence. a8 = 75 a20 = 39 an = a1 + d(n – 1) 39 75 20 8 3 75 = a1 - 3(8 – 1) a1 = 96 d = -3 an = an - 1 - 3, a1 = 96 DG15 --- 10 minutes Find the sum of a Finite Arithmetic Sequence Generate Terms of an Arithmetic Sequence 2 Ways: 1st Way: Repeated addition of d to the initial term Sn a1 a2 a3 ... an2 an1 an a1 a1 d a1 2d ... a1 n 1 d 2nd way: Repeated subtraction of d from the nth term Sn an an1 an2 ... a3 a2 a1 an an d an 2d ... an n 1d RECALL: How can you solve systems of equations? Graphing 1. _________________________ Substitution 2. ________________________ We will use elimination to Elimination 3. ________________________ derive the sum formula. Add these two versions of Sn. Sn a1 a1 d a1 2d ... a1 n 1d Sn an an d an 2d ... an n 1d a1 dn d an dn d n a1 an a1 and an are both added “n” times 2 Ex 3. Find the sum of the odd integers 1 to 19. The formula says we need1 to know n, the first term, and the nth term. 19 Since we are asked to find the sum of the first 10 10 terms, n = ______ n a1 an 101 19 Sn 100 2 2 Ex. 4 Find S20 for the sequence a n 6n 1 20 6(1) – 1 = 6 – 1 = 5 6(20) – 1 = 120 – 1 = 119 S20 205 119 1240 2 • Finding the nth partial sum of an infinite arithmetic sequence--- sum of the first n terms of an infinite sequence Ex 5. Find the 150th partial sum of the arithmetic sequence 5, 16, 27, 38, 49, … an 11n 6 1644 1505 1644 Sn 123,675 2 Ex. 6 An auditorium has 20 rows of seats. There are 20 seats in the first row, 21 seats in the second row, 22 seats in the third row, and so on. How many tickets need to be sold to sell out this auditorium for an upcoming show? To find the sum of all 20 rows, we need: 1 20 20 n = ____ d = ___ and a1 = _____ What is the explicit formula for this problem? an 20 1(n 1) n 19 Find the 20th term: a20 19 20 39 Find the sum of the rows: 2020 39 S 20 590 2 ASSIGNMENT: Textbook p. 635 odds #1 – 43, 59 – 71, 83 – 87 Thurs--- Test Unit 3: Matrices Fri --- DG 15 Unit 4 L1 & L2