Geometric Sequence & Series.pptx
- 1. GEOMETRIC SEQUENCE & SERIES MR. REGIE R. NAUNGAYAN TEACHER III DEPED BANAYOYO NATIONAL HIGH SCHOOL
- 2. OBJECTIVES After going through this module, the learner should be able to: a. determine a geometric sequence; b. identify the common ratio of a geometric sequence; c. find the missing term of a geometric sequence; and, d. determine whether a sequence is geometric or arithmetic. e. insert geometric means in a sequence f. solve the sum of a geometric sequence.
- 3. WHAT IS A GEOMETRIC SEQUENCE? A geometric sequence is a sequence obtained by multiplying a common ratio to the preceding terms in order to obtain the succeeding terms.
- 4. HOW TO DETERMINE THE COMMON RATIO? The common ratio is obtained by dividing a term by the term preceding it. 𝑎2 𝑎1 , 𝑎3 𝑎2 , 𝑎4 𝑎3 , … ࢙࢛ࢉࢉࢋࢋࢊࢍ ࢚ࢋ࢘ ࢘ࢋࢉࢋࢊࢍ ࢚ࢋ࢘
- 5. EXAMPLES 1, 2, 4, 8, 16, … 3, -6, 12,-24, … 3, 1, 1 3 , 1 9 , 1 27 … 1 3
- 6. ACTIVITY 1: I’LL TELL YOU WHAT YOU ARE State whether each of the following sequences is geometric or not. 1. 5, 20, 80, 320, … 2. 7 2, 5 2, 3 2, 2, … 3. 5, -10, 20, -40, … 4. 1, 0.6, 0.36, 0.216 5. 10/3, 10/6, 10/9, 10/15, … Geometric Not Geometric Geometric Geometric Not Geometric
- 7. Identify which of the following is a geometric sequence. If observed as a geometric sequence, determine r. 1. 1, 2, 4, 12, 16, … 6. 1, 3, 9, 27, 81, . . . 2. -2, 4, -8, 16, -32, … 7. 27, 9, 3, 1, 1/3, … 3. 10, 20, 30, 40, 50… 8. 3, 12, 48, 192, 768, . . . 4. 5, 10, 20, 40, 80, … 9. 2, 26, 338, 4 394, … 5. 4, 12, 36, 108, 324 … 10. 5x2, 5x5, 5x8, 5x11, 5x14, … X r = -2 X r = 2 r = 3 r = 3 r = 1/3 r = 4 r = 13 r = 𝒙𝟑 ACTIVITY 2: GEOMETRIC BA ‘YAN?
- 9. ACTIVITY 3: SHADE THAT BOX WORKSHEET Shade the box with blue if the sequence inside it is geometric, but color it red if it contains an arithmetic sequence. Leave the box uncolored if neither a geometric nor arithmetic.
- 10. ACTIVITY 3: SHADE THAT BOX (ANSWER)
- 13. EXAMPLE Find the 9th term of the sequence 1, 3, 9, 27, … a1 = 1 r = 3 n = 9 an = a1rn-1 a9 = (1)(3)9-1 a9 = (1)(3)8 a9 = (1)(6561) a9 = 6561
- 14. EXAMPLE Find the 10th term of the sequence -2, 8, -32, 128, … a1 = -2 r = -4 n = 10 an = a1rn-1 a10 = (-2)(-4)10-1 a10 = (-2)(-4)9 a10 = (-2)(-262,144) a10 = 524,288
- 15. EXAMPLE What is the 5th term of the geometric sequence whose a1 = 1 5 , and r = 2? an = a1rn-1 a5 = 1 5 (2)5-1 a5 = 1 5 (2)4 a5 = 1 5 (16) a5 = 16 5
- 16. TRY THESE! 1. What is the 5th term of the geometric sequence whose a1 = 100, and r = - 1 2 ? 2. Find a7 of the geometric sequence with a1 = −7, and r = 3? a7 = -5,103 a5 = 𝟐𝟓 𝟒
- 17. GEOMETRIC MEAN
- 18. ACTIVITY 4: WHAT’S THAT TERM? Find the specified term in each of the following geometric sequence. 1. 3, 9, 27, … a6 2. 1, -2, 4, -8, … a7 3. 12, a2, 3 4. 2, a2, a3, 54 a6 = 729 a7 = 64 a2 = 6 a2 = 6, a2 = 18
- 19. DEFINITION The given terms are the first and last terms. These terms are called the extremes, and the term/s in between the extremes are called geometric mean/s.
- 20. DEFINITION 2, 4, 8, 16, 32
- 21. FORMULA To insert terms, let us identify first the common ratio by using the formula: 𝒓 = 𝒏−𝟏 𝒂𝒏 𝒂𝟏
- 22. EXAMPLE Find the geometric mean of 4 and 484. 𝒓 = 𝒏−𝟏 𝒂𝒏 𝒂𝟏 𝒓 = 𝟑−𝟏 𝟒𝟖𝟒 𝟒 𝒓 = 𝟐 𝟏𝟐𝟏 𝒓 = ± 11 4, ___, 484 a1 an n = 3
- 23. EXAMPLE Find the geometric mean of 4 and 484. If 𝑟 = 11, then the sequence is 4, 44, 484, which means that the geometric mean is 44. If 𝑟 = -11, then the sequence is 4, -44, 484, which means that the geometric mean is - 44. 𝑟 = ±11 denotes two possible answers
- 24. EXAMPLE Insert two geometric means between 6 and 750. 𝒓 = 𝒏−𝟏 𝒂𝒏 𝒂𝟏 𝒓 = 𝟒−𝟏 𝟕𝟓𝟎 𝟔 𝒓 = 𝟑 𝟏𝟐𝟓 𝒓 = 5 6, ___, ___, 750 a1 an n = 4 If r = 5, then the geometric means are 30 and 150. 30 150
- 25. EXAMPLE Insert two geometric means between 5 184 and 3. 𝒓 = 𝒏−𝟏 𝒂𝒏 𝒂𝟏 𝒓 = 𝟒−𝟏 𝟑 𝟓𝟏𝟖𝟒 𝒓 = 𝟑 𝟏 𝟏𝟕𝟐𝟖 𝒓 = 𝟏 𝟏𝟐 5184, ___, ___, 3 a1 an n = 4 43 2 36
- 26. YOUR TURN! 1. Insert three geometric means between 7 and 16 807. 2. Insert four geometric means between 5 and 38 880. 30, 180, 1080, 6480 49, 343, 2401 or -49, 343, -2401
- 27. GEOMETRIC SERIES
- 28. FORMULA Sn = a1(1 −rn) 1 −r
- 29. EXAMPLE Find the sum of the first 6 terms of a sequence with a1 = 4, r = 5. a1 = 4 r = 5 n = 6 Sn = a1(1 −rn) 1 −r S6 = 4 (1 −(5)6) 1 − 5 S6 = 4 (1 − 15 625) −4 S6 = 4 (−15 624) −4 S6 = −62 496 −4 S6 = 15 624
- 30. EXAMPLE Find the sum of the first 12 terms of the geometric sequence 3, -9, 27, -81, 243, … a1 = 3 r = -3 n = 12 Sn = a1(1 −rn) 1 −r S12 = 3 (1 −(−3)12) 1 −(−3) S12 = 3 (1 −531 441) 1+3 S12 = 3 (−531 440) 4 S12 = −1 594 320 4 S12 = -398 580
- 31. EXAMPLE If a1 = 318, r = 1 2 , what is the sum of the first 7 terms? a1 = 318 r = 1 2 n = 7 Sn = a1(1 −rn) 1 −r S7 = 318 1 − 1 2 7 1 − 1 2 S7 = 318 1 − 1 128 1 2 S7 = 318 127 128 1 2 S7 = 318 127 128 2 1 S7 = 20 193 32
- 32. Find the sum of the indicated series. 1. a1 = 4, r = 2, n = 20 2. a1 = -5, r = 3, n = 9 3. a1 = 216, r = 1 2 , n = 5 4. -6, 24, -96, 384, … n = 7 5. 1.6, 0.08, 0.004, … n = 5 ACTIVITY 5: SUM IT UP!