An Introduction to Sequences & Series

1. An Introduction to Sequences & Series p. 651

2. Sequence: • A list of ordered numbers separated by commas. • Each number in the list is called a term. • For Example: Sequence 1Sequence 2 2,4,6,8,10 2,4,6,8,10,… Term 1, 2, 3, 4, 5 Term 1, 2, 3, 4, 5 Domain – relative position of each term (1,2,3,4,5) Usually begins with position 1 unless otherwise stated. Range – the actual terms of the sequence (2,4,6,8,10)

3. Sequence 1Sequence 2 2,4,6,8,10 2,4,6,8,10,… A sequence can be finite or infinite. The sequence has a last term or final term. (such as seq. 1) The sequence continues without stopping. (such as seq. 2) Both sequences have a general rule: an = 2n where n is the term # and an is the nth term. The general rule can also be written in function notation: f(n) = 2n

4. Write the first 6 terms of an=5-n. a1=5-1=4 a2=5-2=3 a3=5-3=2 a4=5-4=1 a5=5-5=0 a6=5-6=-1 4,3,2,1,0,-1 Write the first 6 terms of an=2n. a1=21=2 a2=22=4 a3=23=8 a4=24=16 a5=25=32 a6=26=64 2,4,8,16,32,64 Examples:

5. The seq. can be written as: Or, an=2/(5n) The seq. can be written as: 2(1)+1, 2(2)+1, 2(3)+1, 2(4)+1,… Or, an=2n+1 Examples: Write a rule for the nth term.

6. Example: write a rule for the nth term. • 2,6,12,20,… • Can be written as: 1(2), 2(3), 3(4), 4(5),… Or, an=n(n+1)

7. Graphing a Sequence • Think of a sequence as ordered pairs for graphing. (n , an) • For example: 3,6,9,12,15 would be the ordered pairs (1,3), (2,6), (3,9), (4,12), (5,15) graphed like points in a scatter plot * Sometimes it helps to find the rule first when you are not given every term in a finite sequence. Term # Actual term

8. Series • The sum of the terms in a sequence. • Can be finite or infinite • For Example: Finite Seq.Infinite Seq. 2,4,6,8,10 2,4,6,8,10,… Finite SeriesInfinite Series 2+4+6+8+10 2+4+6+8+10+…

9. Summation Notation • Also called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series 2+4+6+8+10 can be written as: i is called the index of summation (it’s just like the n used earlier). Sometimes you will see an n or k here instead of i. The notation is read: “the sum from i=1 to 5 of 2i” i goes from 1 to 5.

10. Summation Notation for an Infinite Series • Summation notation for the infinite series: 2+4+6+8+10+… would be written as: Because the series is infinite, you must use i from 1 to infinity (∞) instead of stopping at the 5th term like before.

11. a. 4+8+12+…+100 Notice the series can be written as: 4(1)+4(2)+4(3)+…+4(25) Or 4(i) where i goes from 1 to 25. Notice the series can be written as: Examples: Write each series in summation notation.

12. Example: Find the sum of the series. • k goes from 5 to 10. • (52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1) = 26+37+50+65+82+101 = 361

13. Special Formulas (shortcuts!)

14. Example: Find the sum. • Use the 3rd shortcut!

15. Arithmetic Sequences & Series

16. Arithmetic Sequence: • The difference between consecutive terms is constant (or the same). • The constant difference is also known as the common difference (d). (It’s also that number that you are adding everytime!)

17. -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.

18. Rule for an Arithmetic Sequence an=a1+(n-1)d

19. Example: Write a rule for the nth term of the sequence 32,47,62,77,… . Then, find a12. • The 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

20. 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

21. 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.

22. 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

23. 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

24. 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

25. 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+… .

26. -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!

27. Geometric Sequences & Series

28. Geometric Sequence • The ratio of a term to it’s previous term is constant. • This means you multiply by the same number to get each term. • This number that you multiply by is called the common ratio (r).

29. 4,-8,16,-32,… -8/4=-2 16/-8=-2 -32/16=-2 Geometric (common ratio is -2) 3,9,-27,-81,243,… 9/3=3 -27/9=-3 -81/-27=3 243/-81=-3 Not geometric Example: Decide whether each sequence is geometric.

30. Rule for a Geometric Sequence an=a1rn-1 • Example: Write a rule for the nth term of the sequence 5, 2, 0.8, 0.32,… . Then find a8. • First, find r. • r= 2/5 = .4 • an=5(.4)n-1 a8=5(.4)8-1 a8=5(.4)7 a8=5(.0016384) a8=.008192

31. If a4=3, then when n=4, an=3. Use an=a1rn-1 3=a1(3)4-1 3=a1(3)3 3=a1(27) 1/9=a1 an=a1rn-1 an=(1/9)(3)n-1 To graph, graph the points of the form (n,an). Such as, (1,1/9), (2,1/3), (3,1), (4,3),… Example: One term of a geometric sequence is a4=3. The common ratio is r=3. Write a rule for the nth term. Then graph the sequence.

32. Example: Two terms of a geometric sequence are a2=-4 and a6=-1024. Write a rule for the nth term. • Write 2 equations, one for each given term. a2=a1r2-1 OR -4=a1r a6=a1r6-1 OR -1024=a1r5 • Use these 2 equations & substitution to solve for a1 & r. -4/r=a1 -1024=(-4/r)r5 -1024=-4r4 256=r4 4=r & -4=r If r=4, then a1=-1. an=(-1)(4)n-1 If r=-4, then a1=1. an=(1)(-4)n-1 an=(-4)n-1 Both Work!

33. Formula for the Sum of a Finite Geometric Series n = # of terms a1 = 1st term r = common ratio

34. Find the sum of the first 10 terms. Find n such that Sn=31/4. Example: Consider the geometric series 4+2+1+½+… .

35. log232=n

36. p.675 Infinite Geometric Series

37. The sum of an infinite geometric series

38. Example: Find the sum of the infinite geometric series. For this series, a1=2 & r=0.1

39. Example: Find the sum of the series: So, a1=12 and r=1/3 S=18

40. Example: An infinite geom. Series has a1=4 & a sum of 10. What is the common ratio? 10(1-r)=4 1-r = 2/5 -r = -3/5

41. Example: Write 0.181818… as a fraction. 0.181818…=18(.01)+18(.01)2+18(.01)3+… Now use the rule for the sum!