geometric-sequence-means-and -seriees.ppt
- 1. Find the ratio. 1. 5, 15 2. -10, 20 3. 4, -16 4. -9, -45 5. 10, 30 6. -54, 9 7. -16, -8 8. 27, -9 9. -33, -11 r = 3 r = -2 r = -4 r = 5 r = 3 r = -1/6 r = ½ r = -1/3 r = 1/3
- 3. GeometricSequence • What if your pay check started at $100 a week and doubled every week. What would your salary be after four weeks?
- 4. GeometricSequence • Starting $100. • After one week - $200 • After two weeks - $400 • After three weeks - $800 • After four weeks - $1600. • These values form a geometric sequence.
- 5. Geometric Sequence • Geometric Sequence: a sequence in which each term after the first is found by multiplying the previous term by a constant value called the common ratio.
- 6. Geometric Sequence • Find the common ratio of the sequence 2, -4, 8, -16, 32, … • To find the common ratio, divide any term by the previous term. • 8 ÷ -4 = -2 • r = -2
- 7. Identifying Geo. seq 5, 20, 80, 320, … r = 4 5, -10, 20, -40, … r = -2 4, 0, 0, 0, 0, … r = 0
- 8. Determine whether each sequence is arithmetic, geometric. If the sequence is arithmetic, give the common difference; if geometric, give the common ratio. 1. 6, 18, 54, 162, ... 2. 4, 10, 16, 22, ... 3. 625, 125, 25, 5, … 4. –1296, 216, –36, 6 5. 8.2, 8, 7.8, 7.6, ... 6. 11, 2, -7, -16, ... Geometric r=3 Arithmetic d=6 Geometric r=1/5 Geometric r=-1/6 Arithmetic d=-0.2 Arithmetic d=-9
- 9. Find the common ratio and fill in the blanks 1. 3, 12, 48, __, __ 2. __, __, 32, 64, 128, ... 3. 120, -60, 30, __, __, __ 4. 5, __, 20, 40, __, __ 5. 4, 12, 36, __, __ 6. –2, __, __, –16 –32 –64 7. 256, __, __, –32, 16, ... 8. 27, 9, __, __, 1/3 9. 1/4 __, __, __, 64, 256
- 10. Geometric Sequence • Find the first five terms of the geometric sequence with a1 = -3 and common ratio (r) of 5. • -3, -15, -75, -375, -1875
- 11. Geometric Sequence • Just like arithmetic sequences, there is a formula for finding any given term in a geometric sequence.
- 12. Examples • Our formula for finding any term of a geometric sequence is an = a1(rn-1 )
- 13. Find the nth term 3, 12, 48, … 10th -2, -4, -8, -16, … 12th
- 14. Hw#3: Find the nth term 27, 9, 3, … 15th
- 15. Geometric Means
- 16. Examples • -64, -16, -4, -1, -1/4 • Just like with arithmetic sequences, the missing terms between two nonconsecutive terms in a geometric sequence are called geometric means.
- 17. Geometric Means • Looking at the geometric sequence 3, 12, 48, 192, 768 the geometric means between 3 and 768 are 12, 48, and 192. • Find two geometric means between -5 and 625.
- 18. Geometric Means • -5, __, __, 625 • We need to know the common ratio. Since we only know nonconsecutive terms we will have to use the formula and work backwards.
- 19. Geometric Means • -5, __, __, 625 • 625 is a4, -5 is a1. • 625 = -5•r4-1 divide by -5 • -125 = r3 take the cube root of both sides • -5 = r
- 20. Geometric Means • -5, __, __, 625 • Now we just need to multiply by -5 to find the means. • -5 • -5 = 25 • -5, 25, __, 625 • 25 • -5 = -125 • -5, 25, -125, 625
- 21. Geometric Series
- 22. Geometric Series • Geometric Series - the sum of the terms of a geometric sequence. • Geo. Sequence: 1, 3, 9, 27, 81 • Geo. Series: 1+3 + 9 + 27 + 81 • What is the sum of the geometric series?
- 23. Geometric Series • 1 + 3 + 9 + 27 + 81 = 121 • The formula for the sum Sn of the first n terms of a geometric series is given by Sn= 1- r a1- a1rn or Sn= 1- r a1(1- rn )