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  • 1. Geometric Sequences
  • 2. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.
  • 3. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.
  • 4. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.
  • 5. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.The converse of this fact is also true.
  • 6. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.The converse of this fact is also true. Below is the formulathat we used for working with geometric sequences.
  • 7. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.The converse of this fact is also true.For example, from the sequence above we see that16/8 = 8/4 = 4/2 = 2 = ratio r. Below is the formula that weused for working with geometric sequences.
  • 8. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.The converse of this fact is also true. Below is the formulathat we used for working with geometric sequences.For example, from the sequence above we see that16/8 = 8/4 = 4/2 = 2 = ratio rTheorem: If a1, a2 , a3 , …an is a sequence such thatan+1 / an = r for all n, then a1, a2, a3,… is a geometricsequence and an = a1*rn-1.
  • 9. Geometric SequencesA sequence a1, a2 , a3 , … is a geometric sequence ifan = crn, i.e. it is defined by an exponential formula.The sequence of powers of 2a1= 2, a2= 4, a3= 8, a4= 16, …is a geometric sequence because an = 2n.Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, thenthe ratio between any two consecutive terms is r.The converse of this fact is also true. Below is the formulathat we used for working with geometric sequences.For example, from the sequence above we see that16/8 = 8/4 = 4/2 = 2 = ratio rTheorem: If a1, a2 , a3 , …an is a sequence such thatan+1 / an = r for all n, then a1, a2, a3,… is a geometricsequence and an = a1*rn-1. This is the general formula forgeometric sequences.
  • 10. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.
  • 11. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6
  • 12. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18
  • 13. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
  • 14. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1
  • 15. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)
  • 16. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)If a1, a2 , a3 , …an is a geometric sequence such that the termsalternate between positive and negative signs,then r is negative.
  • 17. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)If a1, a2 , a3 , …an is a geometric sequence such that the termsalternate between positive and negative signs,then r is negative.Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometricsequence because-1/(2/3) = (3/2) / (-1)
  • 18. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)If a1, a2 , a3 , …an is a geometric sequence such that the termsalternate between positive and negative signs,then r is negative.Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometricsequence because-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2)
  • 19. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)If a1, a2 , a3 , …an is a geometric sequence such that the termsalternate between positive and negative signs,then r is negative.Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometricsequence because-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.
  • 20. Geometric SequencesGiven the description of a geometric sequence, we use thegeneral formula to find the specific formula for that sequence.Example A. The sequence 2, 6, 18, 54, … is a geometricsequence because 6/2 = 18/6 = 54/18 = … = 3 = r.Since a1 = 2, set them in the general formula of the geometricsequences an = a1r n – 1 , we get the specific formula for thissequence an = 2*3(n – 1)If a1, a2 , a3 , …an is a geometric sequence such that the termsalternate between positive and negative signs,then r is negative.Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometricsequence because-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.Since a1 = 2/3, the specific formula is 2 –3 n–1 an = 3 ( 2 )
  • 21. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.
  • 22. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,
  • 23. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1
  • 24. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1)
  • 25. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12
  • 26. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12
  • 27. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12
  • 28. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾
  • 29. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾b. find the specific equation.
  • 30. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾b. find the specific equation.Set a1 = ¾ and r = -2 into the general formula an = a1rn – 1 ,
  • 31. Geometric SequencesTo use the geometric general formula to find the specificformula, we need the first term a1 and the ratio r.Example C. Given that a1, a2 , a3 , …is a geometric sequencewith r = -2 and a5 = 12,a. find a1By that the general geometric formulaan = a1r n – 1, we geta5 = a1(-2)(5 – 1) = 12 a1(-2)4 = 12 16a1 = 12 a1 = 12/16 = ¾b. find the specific equation.Set a1 = ¾ and r = -2 into the general formula an = a1rn – 1 ,we get the specific formula of this sequence an= 3 (-2)n–1 4
  • 32. Geometric SequencesC. Find a9.
  • 33. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4
  • 34. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4
  • 35. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3a9 = 4 (-2)8
  • 36. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192
  • 37. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,
  • 38. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1
  • 39. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,
  • 40. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,we have a3 = -2 = a1r3–1
  • 41. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1
  • 42. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2
  • 43. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2 54 = a1r5
  • 44. Geometric SequencesC. Find a9.Since an= 3 (-2)n–1, 4set n = 9, we geta9= 3 (-2)9–1 4 3 3a9 = 4 (-2)8 = 4 (256) = 192Example D. Given that a1, a2 , a3 , …is a geometric sequencewith a3 = -2 and a6 = 54,a. Find r and a1Given that the general geometric formula an = a1rn – 1,we have a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1 -2 = a1r2 54 = a1r5Divide these equations:
  • 45. Geometric Sequences54 a1r5 = a r2-2 1
  • 46. Geometric Sequences-27 54 = a1r 5 -2 a1r2
  • 47. Geometric Sequences-27 54 = a1r 5 -2 a1r2
  • 48. Geometric Sequences 3 = 5-2-27 54 = a1r 5 -2 a1r2
  • 49. Geometric Sequences 3 = 5-2-27 54 = a1r 5 -2 a1r2 -27 = r3
  • 50. Geometric Sequences 3 = 5-2-27 54 = a1r 5 -2 a1r2 -27 = r3 -3 = r
  • 51. Geometric Sequences 5 3 = 5-2-27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2
  • 52. Geometric Sequences 5 3 = 5-2-27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2
  • 53. Geometric Sequences 5 3 = 5-2-27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19
  • 54. Geometric Sequences 5 3 = 5-2-27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1
  • 55. Geometric Sequences 5 3 = 5-2-27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2
  • 56. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3
  • 57. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3 we have the specific formulaan = -2 (-3)n–1 9
  • 58. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3 we have the specific formulaan = -2 (-3)n–1 9To find a2, set n = 2, we geta2 = -2 (-3) 2–1 9
  • 59. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3 we have the specific formulaan = -2 (-3)n–1 9To find a2, set n = 2, we geta2 = -2 (-3) = -2 (-3) 2–1 9 9
  • 60. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3 we have the specific formulaan = -2 (-3)n–1 9To find a2, set n = 2, we geta2 = -2 (-3) = -2 3 (-3) 2–1 9 9
  • 61. Geometric Sequences 5 3 = 5-2 -27 54 = a1r -2 a1r2 -27 = r3 -3 = rPut r = -3 into the equation -2 = a1r2Hence -2 = a1(-3)2 -2 = a19 -2/9 = a1b. Find the specific formula and a2Use the general geometric formula an = a1rn – 1,set a1 = -2/9, and r = -3 we have the specific formulaan = -2 (-3)n–1 9To find a2, set n = 2, we geta2 = -2 (-3) = -2 3 (-3) = 2 2–1 9 9 3
  • 62. Geometric Sequences
  • 63. Geometric SequencesSum of Geometric Sequences
  • 64. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2
  • 65. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3
  • 66. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4
  • 67. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
  • 68. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
  • 69. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rnHence 1 + r + r + … + r = 2 n-1 1 – rn 1–r
  • 70. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rnHence 1 + r + r + … + r = 2 n-1 1 – rn 1–rTherefore a1 + a1r + a1r2 + … +a1rn-1
  • 71. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rnHence 1 + r + r + … + r = 2 n-1 1 – rn 1–rTherefore a1 + a1r + a1r2 + … +a1rn-1 n terms
  • 72. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rnHence 1 + r + r + … + r = 2 n-1 1 – rn 1–rTherefore a1 + a1r + a1r2 + … +a1rn-1 n terms= a1(1 + r + r2 + … + r n-1)
  • 73. Geometric SequencesSum of Geometric SequencesWe observe the algebraic patterns:(1 – r)(1 + r) = 1 – r2(1 – r)(1 + r + r2) = 1 – r3(1 – r)(1 + r + r2 + r3) = 1 – r4(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5...…(1 – r)(1 + r + r2 … + rn-1) = 1 – rnHence 1 + r + r + … + r = 2 n-1 1 – rn 1–rTherefore a1 + a1r + a1r2 + … +a1rn-1 n terms a1 1 – r n= a1(1 + r + r + … + r ) = 2 n-1 1–r
  • 74. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–r
  • 75. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)
  • 76. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2,
  • 77. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16.
  • 78. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms.
  • 79. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2
  • 80. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 16
  • 81. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2
  • 82. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2 -243 = ( - 3 ) n – 1 32 2
  • 83. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2 -243 = ( - 3 ) n – 1 32 2Compare the denominators we see that 32 = 2n – 1.
  • 84. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2 -243 = ( - 3 ) n – 1 32 2Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1
  • 85. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2 -243 = ( - 3 ) n – 1 32 2Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1 n–1=5
  • 86. Geometric SequencesFormula for the Sum Geometric Sequencesa1 + a1r + a1r + … +a1r = a1 2 n-1 1 – rn 1–rExample E. Find the geometric sum :2/3 + (-1) + 3/2 + … + (-81/16)We have a1 = 2/3 and r = -3/2, and an = -81/16. We need thenumber of terms. Put a1 and r in the general formula we getthe specific formulaan= 2 ( - 3 ) n-1 3 2To find n, set an = -81 = 2 ( - 3 ) n – 1 16 3 2 -243 = ( - 3 ) n – 1 32 2Compare the denominators we see that 32 = 2n – 1.Since 32 = 25 = 2n – 1 n–1=5 n=6
  • 87. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)
  • 88. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–r
  • 89. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2)
  • 90. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2) = 2 1 – (729/64) 3 1 + (3/2)
  • 91. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2) = 2 1 – (729/64) 3 1 + (3/2) = 2 -665/64 3 5/2
  • 92. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2) = 2 1 – (729/64) 3 1 + (3/2) = 2 -665/64 3 5/2 = -133 48
  • 93. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2) = 2 1 – (729/64) 3 1 + (3/2) = 2 -665/64 3 5/2 = -133 48
  • 94. Geometric SequencesTherefore there are 6 terms in the sum,2/3 + (-1) + 3/2 + … + (-81/16)Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – rn 1–rwe get the sum S 2 1 – (-3/2)6S = 3 1 – (-3/2) = 2 1 – (729/64) 3 1 + (3/2) = 2 -665/64 3 5/2 = -133 48