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CHAPTER 1 SEQUENCES AND SERIES MALAYSIA INSTITUTE OF INFORMATION TECHNOLOGY UNIVERSITI KUALA LUMPUR UPDATED: JUN 2011
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1.1 SEQUENCE <ul><li>Consider the following sequence: </li></ul><ul><li>Sequence 1 Sequence 2 </li></ul><ul><li>2, 4, 6, 8, 10 2, 4, 6, 8, 10….. </li></ul>1 st term 2nd term 3 rd term 4 th term 5 th term 1 st term 2nd term 3 rd term 4 th term 5 th term Each no in the list is called a term . Each term are separated by commas . Finite sequence end after a certain no of terms. Infinite sequence is one that continues indefinitely (non stop). finite sequence infinite sequence
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1.1.1 Finding the formula for the n th terms of a sequence <ul><li>Example 1: </li></ul><ul><li>Find the n th term of the following sequence </li></ul><ul><li>Solution: Solution: </li></ul><ul><li>The seq. can be written as The seq. can be written as </li></ul><ul><li>The n th term is The n th term is </li></ul>The formula for the terms in sequence can be given as a formula for the n th term. 2/(5 n ) 2(n)+1
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<ul><li>Solution: </li></ul><ul><li>The seq. can be written as </li></ul><ul><li>The n th term is </li></ul><ul><li>Example 2: Example 3: </li></ul><ul><li>Write the first 6 terms of x(k)=5-k. </li></ul><ul><li>Solution: </li></ul><ul><li>X(1)=5-1=4 </li></ul><ul><li>X(2)=5-2=3 </li></ul><ul><li>X(3)=5-3=2 </li></ul><ul><li>X(4)=5-4=1 </li></ul><ul><li>X(5)=5-5=0 </li></ul><ul><li>X(6)=5-6=-1 </li></ul><ul><li>The first 6 terms is </li></ul><ul><li>4,3,2,1,0,-1 </li></ul>Write the first 6 terms of x(k)=2 k . Solution: The first 6 terms is 2,4,8,16,32,64
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1.2 SERIES <ul><li>Consider the following sequence: </li></ul><ul><li>Series 1 Series 2 </li></ul><ul><li>2 + 4 + 6 + 8 +10 2 + 4 + 6 + 8 + 10….. </li></ul><ul><li>Example: </li></ul><ul><li>Finite series </li></ul><ul><li>Infinite series </li></ul>1 st term 2nd term 3 rd term 4 th term 5 th term 1 st term 2nd term 3 rd term 4 th term 5 th term Series is the sum of terms in a sequence. Summation notation is Finite series end after a certain no of terms. Infinite series is one that continues indefinitely (non stop). finite series infinite series Last value of r in the sequence First value of r in the sequence First value of r in the sequence Infinity
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<ul><li>Example 4: </li></ul><ul><li>Write the terms of the series. </li></ul><ul><li>Solution: Solution: </li></ul><ul><li>Example 5: </li></ul><ul><li>Write in the summation notation </li></ul><ul><li>Solution: Solution: </li></ul><ul><li>The series can be written as: The series can be written as: </li></ul>
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<ul><li>Example 6: </li></ul><ul><li>Determine the sum of the following series. </li></ul><ul><li>Solution: </li></ul><ul><li>Solution: </li></ul>
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1. 3 ARITHMETIC SEQUENCE AND SERIES 1.3.1 ARITHMETIC SEQUENCE <ul><li>The difference between consecutive terms is same. </li></ul><ul><li>The difference is called common difference, d. </li></ul><ul><li>Consider the following sequence </li></ul><ul><li>Each terms has 4 added to it to obtain the next term. </li></ul><ul><li>The sequence is said to have a common difference of 4 (d = 4). </li></ul>14 -10 6 -2 10 - 6 d = 4 d = 4 d = 4
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Formula for the n th term, a = the first term d = common difference Example 7: Find the tenth term of the arithmetic sequence: 32, 47, 62, 77,…..
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<ul><li>Example 8: </li></ul><ul><li>Find the number of terms in the arithmetic sequence: 50, 47, 44, …., -34 </li></ul>
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1.3.2 ARITHMETIC SERIES <ul><li>A series is the sum of a sequence </li></ul><ul><li>Formula for sum of an arithmetic series, a = the first term d = common difference l = the last term </li></ul><ul><li>Example 9: </li></ul><ul><li>Determine the sum of the first 25 terms of arithmetic series: </li></ul>
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Exercises <ul><li>Find the 23 rd term of an arithmetic sequence with first term 2 and common difference 7. (156) </li></ul><ul><li>Find the sum of the first five terms of the arithmetic sequence with first term 3 and common difference 5. (65) </li></ul><ul><li>Write down the 10 th and 19 th terms of the arithmetic sequence </li></ul><ul><li>a. 8, 11, 14, … (35, 62) </li></ul><ul><li>b. 8, 5, 2, … (-19, -46) </li></ul><ul><li>An arithmetic sequence is given by </li></ul><ul><li>a. State the sixth term (-2b/3) </li></ul><ul><li>b. State the k th term [b(4-k)]/3 </li></ul><ul><li>c. If the 20 th term has a value of 15, find b . -45/16 </li></ul>
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Application Problems <ul><li>After knee surgery, your trainer tells you to return to your jogging program slowly. He suggests jogging for 12 minutes each day for the first week. Each week thereafter, he suggests that you increase that time by 6 minutes per day. How many weeks will it be before you are up to jogging 60 minutes per day? </li></ul><ul><li>You visit the Grand Canyon and drop a penny off the edge of a cliff. The distance the penny will fall is 16 feet the first second, 48 feet the next second, 80 feet the third second, and so on in an arithmetic sequence. What is the total distance the object will fall in 6 seconds? </li></ul>
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Application Problems <ul><li>A shop assistant is arranging a display of a triangular array of tins so as to have one tin in the top row, two in the second, three in the third and so on. If there are 66 tins altogether, how many rows can be completed arranged? ( 11 ) </li></ul>4. A restaurant has square tables which seat four people. When two tables are placed together, six people can be seated. If 20 square tables are placed together to form one long table, how many people can be seated? If 1000 square tables are placed together to form one very long table, how many people can be seated?( 42, 2002 )
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1. 4 GEOMETRIC SEQUENCE AND SERIES 1.4.1 GEOMETRIC SEQUENCE <ul><li>The ratio between consecutive terms is same. </li></ul><ul><li>The ratio is called common ratio, r. </li></ul><ul><li>Consider the following sequence </li></ul><ul><li>Starting with 4, and multiplying each term by -2 would generate the above sequence. </li></ul><ul><li>The sequence is said to have a common ratio of -2 (r = -2). </li></ul>-8/4 16/-8 -32/16 r = -2 r = -2 r = -2
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Formula for the n th term, a = the first term r = common ratio Example 11: Determine the eighth term of the geometric sequence: 4, 12, 36,108,…..
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1.4.2 GEOMETRIC SERIES <ul><li>A series is the sum of a sequence </li></ul><ul><li>Formula for the sum of a geometric series, </li></ul><ul><li> a = the first term r = common ratio </li></ul><ul><li>Example 12: </li></ul><ul><li>Determine the sum of the first 10 terms of geometric series: </li></ul>
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<ul><li>Formula for the sum of a Infinite geometric series, </li></ul><ul><li> a = the first term r = common ratio </li></ul><ul><li>Example 12: </li></ul><ul><li>Determine the sum of a Infinite geometric series with the first term is 3 and the common ration is ½ : </li></ul>
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Exercises <ul><li>Find the seventh term of a geometric sequence with first term 2 and common ratio 3. (1458) </li></ul><ul><li>Find the sum of the first five terms of the geometric sequence with the first term 3 and common ratio 2. (93) </li></ul><ul><li>Find the sum of the infinite geometric series with first term 2 and common ratio ½. (4) </li></ul><ul><li>A geometric sequence has first term 1. The ninth term exceed the fifth term by 240. Find possible values for the eighth term. (+/- 128) </li></ul><ul><li>The sum to infinity of a geometric sequence is four times the first term. Find the common ratio. (3/4) </li></ul><ul><li>6. A geometric series has S 3 = 37/8 and S 6 = 367/512. Find the first term and the common ratio. (2, ¾) </li></ul><ul><li>A geometric sequence is given by 1, ½, ¼, … What is its common ratio? (1/2) </li></ul>
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Solution Exercise No 4 <ul><li>A geometric sequence has first term 1. The ninth term exceed the fifth term by 240. Find possible values for the eighth term. (+/- 128) </li></ul>Step 1: Step 2: Step 3:
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Solution Exercise No 6 <ul><li>A geometric series has S 3 = 37/8 and S 6 = 3367/512. Find the first term and the common ratio. (2, ¾ ) </li></ul>Step 1: Step 2: Step 3: Step 4:
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Application problems <ul><li>A culture of bacteria doubles every 2 hours. If there are 500 bacteria at the beginning, how many bacteria will there be after 24 hours? </li></ul><ul><li>Recent estimates, based on data from satellite observations, report 775 million hectares of rain forest remaining. The average annual rate of deforestation in the world is 0.77%. How many million hectares of rain forest will be lost in the next decade? (57.65 millions) </li></ul>
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