7 .2 Analyze Arithmetic Sequences & Series

1. 7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for an arithmetic sequence? How do you find the rule when given two terms?

2. Arithmetic Sequence: • The difference between consecutive terms is constant (or the same). • The constant difference is also known as the common difference (d). Find the common difference by subtracting the term on the left from the next term on the right.

3. -10,-6,-2,0,2,6,10,… -6--10=4 -2--6=4 0--2=2 2-0=2 6-2=4 10-6=4 Not arithmetic (because the differences are not the same) 5,11,17,23,29,… 11-5=6 17-11=6 23-17=6 29-23=6 Arithmetic (commondifference is 6) Example: Decide whether each sequence is arithmetic.

4. Rule for an Arithmetic Sequence

5. Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12. • There is a common difference where d=15, therefore the sequence is arithmetic. • Use an=a1+(n-1)d an=32+(n-1)(15) an=32+15n-15 an=17+15n a12=17+15(12)=197

6. a. One term of an arithmetic sequence is a19 = 48. The common difference is d = 3. a. Write a rule for the nth term. SOLUTION a. Use the general rule to find the first term. an= a1+ (n – 1) d Write general rule. a19 =a1+ (19– 1) d Substitute 19 forn Substitute 48 for a19 and 3 for d. 48 = a1 + 18(3) Solve for a1. – 6 = a1 So, a rule for the nth term is: Write general rule. an= a1+ (n – 1) d = – 6+ (n – 1) 3 Substitute – 6 for a1 and 3 for d. Simplify.

7. b. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. b. Graph the sequence. One term of an arithmetic sequence is a19 = 48. The common difference is d =3.

8. Example: One term of an arithmetic sequence is a8=50. The common difference is 0.25. Write a rule for the nth term. • Use an=a1+(n-1)d to find the 1st term! a8=a1+(8-1)(.25) 50=a1+(7)(.25) 50=a1+1.75 48.25=a1 * Now, use an=a1+(n-1)d to find the rule. an=48.25+(n-1)(.25) an=48.25+.25n-.25 an=48+.25n

9. Now graph an=48+.25n. • Just like yesterday, remember to graph the ordered pairs of the form (n,an) • So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.

10. Example: Two terms of an arithmetic sequence are a5=10 and a30=110. Write a rule for the nth term. • Begin by writing 2 equations; one for each term given. a5=a1+(5-1)d OR 10=a1+4d And a30=a1+(30-1)d OR 110=a1+29d • Now use the 2 equations to solve for a1 & d. 10=a1+4d 110=a1+29d (subtract the equations to cancel a1) -100= -25d So, d=4 and a1=-6 (now find the rule) an=a1+(n-1)d an=-6+(n-1)(4) OR an=-10+4n

11. Example (part 2): using the rule an=-10+4n, write the value of n for which an=-2. -2=-10+4n 8=4n 2=n

12. a27 = a1 + (27– 1)d 97 = a1 + 26d 21 = a1 + 7d a8= a1 + (8– 1)d STEP 2 Solve the system. STEP 1 Write a system of equations using an= a1 +(n – 1)dand substituting 27for n(Eq 1) and then 8for n(Eq 2). STEP 3 Find a rule for an. Two terms of an arithmetic sequence are a8 = 21 and a27 = 97. Find a rule for the nth term. SOLUTION Equation 1 Equation 2 Subtract. 76 = 19d Solve for d. 4 = d Substitute for d in Eq 1. 97 = a1 + 26(4) 27 = a1 Solve for a1. an= a1 + (n – 1)d Write general rule. Substitute for a1 and d. = – 7 + (n – 1)4 = – 11 + 4n Simplify.

13. What is an arithmetic sequence? The difference between consecutive terms is a constant • What is the rule for an arithmetic sequence? an=a1+(n-1)d • How do you find the rule when given two terms? Write two equations with two unknowns and use linear combination to solve for the variables.

14. 7.2 Assignment p. 446, 3-35 odd

15. Analyze Arithmetic Sequences and Series day 2 What is the formula for find the sum of a finite arithmetic series?

16. Arithmetic Series • The sum of the terms in an arithmetic sequence • The formula to find the sum of a finite arithmetic series is: Last Term 1st Term # of terms

17. Find the sum of the 1st 25 terms. First find the rule for the nth term. an=22-2n So, a25 = -28 (last term) Find n such that Sn=-760 Example: Consider the arithmetic series 20+18+16+14+… .

18. -1520=n(20+22-2n) -1520=-2n2+42n 2n2-42n-1520=0 n2-21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive solution!

19. ( ) 8 + 103 2 S20 = 20 The correct answer is C. ANSWER SOLUTION a1= 3 + 5(1) = 8 Identify first term. Identify last term. a20= 3 + 5(20) =103 Write rule for S20, substituting 8 for a1 and 103 for a20. = 1110 Simplify.

20. House Of Cards a. Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12, . . . . These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is: a. Write a rule for the number of cards in the nth row if the top row is row 1. You are making a house of cards similar to the one shown SOLUTION an = a1 + (n– 1) = d Write general rule. Substitute 3 for a1 and 3 for d. = 3 + (n – 1)3 Simplify. = 3n

21. House Of Cards What is the total number of cards if the house of cards has 14 rows? b. You are making a house of cards similar to the one shown SOLUTION Find the sum of an arithmetic series with first term a1 = 3 and last term a14 = 3(14) = 42. b. Total number of cards = S14

22. S12 = 570 ANSWER ( ) a1 + an Sn = n 2 5. Find the sum of the arithmetic series (2 + 7i). i = 1 SOLUTION a1= 2 + 7(1) = 9 a12= 2 + (7)(12) = 2 + 84 = 86 S12 = 570

23. What is the formula for find the sum of a finite arithmetic series?

24. 7.2 Assignment: p. 446 40-48 all, 63-64