Aim: What is an arithmetic sequence and series? Do Now: What is the 10th term of the arithmetic sequence -1, 3, 7, 11, . . .

Do Now Problem What is the 10th term of the arithmetic sequence -1, 3, 7, 11, . . . d = 4 an = a1 + (n – 1)d a10 = -1 + (10 – 1)4 a10 = -1 + (9)5 = 44

A series is the expression for the sum of the terms of a sequence. Arithmetic Series A series is the expression for the sum of the terms of a sequence. finite sequence 6, 9, 12, 15, 18 finite series 6 + 9 + 12 + 15 + 18 a1, a2, a3, a4, a5 a1 + a2 + a3 + a4 + a5 infinite sequence 3, 7, 11, 15, . . . infinite series 3 + 7 + 11 + 15 + . . . a1, a2, a3, a4, . . . a1 + a2 + a3 + a4 + . . .

The Sum of an Arithmetic Sequence: Series When famous German mathematician Karl Gauss was a child, his teacher required the students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class busy for some time. Gauss gave the answer almost immediately. Can you? The sum of a finite arithmetic sequence with n terms is An arithmetic series is the indicated sum of the terms of an arithmetic sequence. Find the following sum 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 Is this an arithmetic sequence? Why? d = 2 10 terms n = 10 a1 = 1 an = 19

The Sum of an Arithmetic Sequence: Series In an arithmetic series, if a1 is the first term, n is the number of terms, an is the nth term, and d is the common difference, then Sn the sum of the arithmetic series, is given by the formulas: or

Model Problem Find the sum of the first ten terms of an arithmetic sequence whose first term is 5 and whose 10th term is -13. a1 = 5 a10 = -13

Model Problem Find the sum of the first fifty terms of an arithmetic sequence 3 + 5 + 7 + 9 + . . . . an = dn + c c = a1 – d an = a1 + (n – 1)d a1 = 5 a50 = ? d = 2, n = 50 an = a1 + (n – 1)d an = 3 + (50 – 1)2 = 101

Model Problem – Option 2 Find the sum of the first fifty terms of an arithmetic sequence 3 + 5 + 7 + 9 + . . . . a1 = 5 d = 2, n = 50

Application A small business sells $10,000 worth of products during its first year. The owner of the business has set a goal of increasing annual sales by $7,500 each year for 9 years. Assuming that this goal is met, find the total sales during the first 10 years this business is in operation. a1 = 10000 d = 7500 c = 10000 – 7500 = 2500 an = 7500n + 2500 a10 = 7500(10)+ 2500 = 77500

Application A man wishes to pay off a debt of $1,160 by making monthly payments in which each payment after the first is $4 more than that of the previous month. According to this plan, how long will it take him to pay the debt if the first payment is $20 and no interest is charged? Sn = 1160 d = 4 a1 = $20 n = ?

Model Problem Sn = 65000 a1 = 3750 d = 500 n = 10.58 and –24.58 Mrs. Gonzales sells houses and makes a commission of $3750 for the first house sold. She will receive a $500 increase in commission for each additional house sold. How many houses must she sell to reach total commissions of $65000? Sn = 65000 a1 = 3750 d = 500 an = a1 + (n – 1)d Navigator: List number of houses sold n = 10.58 and –24.58 She must sell 11 houses.