Applied 40S May 28, 2009

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Geometric sequences.

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Applied 40S May 28, 2009

  1. 1. Sequences are everywhere ... Photo source: Stone throw sequence
  2. 2. The Bug on the Water Wheel A water wheel with a 7.0 ft radius has 1.0 ft. submerged in the water as shown, and rotates counterclockwise at 6.0 revolutions per minute. A bug is sitting on the wheel at point B. You start your stopwatch, and two seconds later the bug at point B is at its greatest height above the water. You are to model the distance 'h' of the bug from the surface of the water in terms of the number of seconds 't' the stopwatch reads. (a) Sketch the graph. (b) Write the algebraic equation of the sinusoid. (c) How far is the bug above the water when t = 5.5 seconds?
  3. 3. Determine which of the following sequences are arithmetic. If a sequence is arithmetic, write the values of a and d. HOMEWORK (a) 5, 9, 13, 17, ... (b) 1, 6, 10, 15, 19, ... a=5 not arithmetic d=4 Given the values of a and d, write the first 5 terms of each arithmetic sequence. (a) a = 7, d, = 2 (b) a = -4, d, = 6 7, 9, 11, 13, 15 -4, 2, 8, 14, 20
  4. 4. List the first 4 terms of the sequence determined by each of the following implicit definitions. HOMEWORK 0, 3, 6, 9 0, 1, 4, 9 1, 2, 4, 8
  5. 5. Sequence: An ordered list of numbers that follow a certain pattern (or rule). Arithmetic Sequence: (i) Recursive Definition: An ordered list of numbers generated by continuously adding a value (the common difference) to a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by a linear equation. Example:
  6. 6. Sequence: An ordered list of numbers that follow a certain pattern (or rule). Common Difference (d): (i) The number that is repeatedly added to successive terms in an arithmetic sequence. (ii) From the implicit definition, d is the slope of the linear equation. Example: 4, 7, 10, 13, , ,
  7. 7. To Find The Common Difference d is the common difference tn is an arbitrary term in the sequence d = tn - t(n - 1) t(n - 1) is the term immediately before tn in the sequence Example: Find the common difference for the sequence: 11, 5, -1, -7, ...
  8. 8. To Find the nth Term In an Arithmetic Sequence t is the nth term n t = a + (n - 1)d a is the first term n n is the quot;rankquot; of the nth term in the sequence d is the common difference Example: Find the 51st term (t51) of the sequence 11, 5, -1, -7, ... Solution: a = 11 t51 = 11 + (51 - 1)(-6) d = 5 - 11 t51 = 11 + (50)(-6) = -6 t51 = 11 - 300 n = 51 t51 = -289
  9. 9. Use your calculator to find the first 10 terms and the sum of the first 10 terms of the sequence: 16, 8, 4, 2, . . . HOMEWORK (a) What is the 10th term? What is the sum of the first 10 terms? (b) Extend the sequence to 15 terms. What is the 15th term? What is the sum of 15 terms? (c) What happens to the terms as you have more terms? Also, what happens to the value of the sum of the terms as you have more terms? (Look at 30, 50, or more terms to verify this answer.)
  10. 10. 16, 8, 4, 2,
  11. 11. 3, 6, 12, 24, , ,
  12. 12. Geometric sequences on the calculator ...
  13. 13. A good resource for learning your way around the calculator or to review what we've learned in class ... Working with Sequences on the TI-83+ or 84+ http://www.mathbits.com/MathBits/TISection/PreCalculus/sequences.htm
  14. 14. Geometic Sequence: (i) Recursive Definition: An ordered list of numbers generated by continuously multiplying a value (the common ratio) with a given first term. (ii) Implicit Definition: An ordered list of numbers where each number in the list is generated by an exponential equation.
  15. 15. Common Ratio (r): (i) The number that is repeatedly multiplied to successive terms in a geometic sequence. (ii) From the implicit definition, r is the base of the exponential function.
  16. 16. To Find The Common Ratio r is the common ratio tn is an arbitrary term in the sequence t(n - 1) is the term immediately before tn in the sequence
  17. 17. To Find the nth Term In a Geometic Sequence tn is the nth term a is the first term n is the quot;rankquot; of the nth term in the sequence r is the common ratio
  18. 18. Once we know the pattern of a sequence, we can find any term of the sequence. Use your calculator to check your answers. Find the 110th term of: 5, 8, 11, 14, ... U(110)= 332 HOMEWORK Find the 21st term of: 3, 6, 12, 24, ... U(21)= 3 145 728
  19. 19. Find the 27th term of: 1, 1, 2, 3, 5, 8, ... HOMEWORK U(21)= 196 418
  20. 20. Which term in the sequence: 2, 5, 11, 23, 47, ... is 1535? HOMEWORK
  21. 21. My cat is sick. His name is Little John and he has a cold. The vet has given him some medicine. Each day he gets a pill with 35 mg of medicine. His body eliminates 25% of the medicine each day and then he gets another pill. HOMEWORK (a) How much medicine will be in his body in 5 days? (b) Will the amount of medicine in his body stabilize? How many days will it take and how much medicine will be in his body?

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