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BA301 : ENGINEERING MATHEMATHICS 3PROGRESSIONS3.1 INTRODUCTION A sequence is a set of terms which are written in a definite order obeying a certain rule. Example of a sequence is 2,4,6,8,.... which is a sequence of even numbers. A series is the sum of the terms of a sequence. Example of a series is 1+ 3 + 5 + 7...3.2 ARITHMETIC PROGRESSION (AP)3.2.1 THE nth TERM OF AN AP An AP is a sequence of numbers where each new term after the first term,a is made by adding on a constant amount to the previous term. This constant is known as the common difference can be positive, zero or negative The common difference,d is given by : The nth term, Tn of an AP is given by : d = Tn + 1 - Tn Tn = a + (n - 1) d Where: Tn + 1 = (n+1) th term Where: a = first term T n = n th term d= common difference An AP can be written as a, a+d, a+2d, a+3d,.... Example 1: a) Calculate the 8th term of an AP: 3, 7, 11, 15, 19, .... 7 b) Calculate the 5th term of an AP: 3, , 4, .... 2 Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 c) Find the 7th and 18th terms of the AP 1.3, 1.1, 0.9, …. Example 2: 1 1 3 a) What is the number of terms in the AP 1 ,2 ,3 ,.....,10 . 4 2 4 b) The 5th term and 6th term of an AP are 27 and 21 respectively. Find the first term. c) Find the nth term of the AP 14, 9, 4, ….. in terms of n. Hence find the 21st term of the AP. d) The 3rd and 5th terms of an AP are -10 and -20 respectively. Find the: i) First term ii) The 12ve term Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 Exercise 1: 1) Find the 6th and the 15th terms of the AP: a. 1, 5, 9, …… b. -4, -7, -10, …… 1 1 2 c. , , ,..... 3 2 3 2) Find the number of terms in the following AP: a. 3, 10, 17, …… , 143 b. -143, -130, …… , 221 5 7 9 33 c. , , ,...., 4 4 4 4 3) Given that the 4th term of an AP is 12 and the common difference is 2. Find the first 3 terms of the AP. 4) Find the number of terms of the AP 407, 401, 395, ….. , -133. Hence, find the 20th term of the AP. 5) Find the nth term of the AP 21, 15, 9, ….. in terms of n. Hence find the 17th term of the AP. 6) The nth term of an AP is given by Tn = 4 – 9n. Find the first term, 10th term and common difference. 7) The 3rd and 12ve terms of an AP are -3 and 24 respectively. Find the: i) First term and common difference ii) The 15th term 3.2.2 SUM OF THE FIRST n TERMS OF AN AP The sum of the first n terms, Sn of an AP is given by: n Sn = [ 2a + ( n - 1 ) d ] 2 Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 Where: a = first term d = common difference Example 3: a) Find the sum of the first 20 terms of the AP – 8, – 3, 2, …. . b) Find the sum of the AP 9.8, 9.2, 8.6, 8.0, …., – 5.2. c) The sequence 20, 26, 32, … is an AP. Find the sum of the 6th term to the 15th term of the AP. d) The sequence 25, 22, 19, …. is an AP. Find the value of n for which the sum of the first n terms of the AP is 116. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 e) The first three terms of an AP are 2p – 1, 4p and 5p + 4. Find: i) The value of p ii) The sum of the first 13 terms of the AP f) The 7th term of an AP is 11 and the sum of the first 16 terms of the AP is 188. Find: i) The first term and the common difference of the AP. ii) The sum of the first 10 terms after the 16th term. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 Exercise 2: 1) Find the sum of : a. The first 9 terms of the AP 7, 11, 15, 19, …. . b. The first 12 terms of the AP 10, 4, -2, …. . 2) Find the sum of the following AP: a. 1 + 3 + 5 + …… + 101 b. 205 + 200 + 195 + 190 + ….. + 105 3) The sequence -30, -22, -14, …. is an AP. Find the sum from the 9th term to the 14th term of the AP. 4) The sequence -9, -5, -1, …. is an AP. Find the value of n for which the sum of the first n terms of the AP is 90. 5) The first three terms of an AP are k – 3, k + 3 and 2k + 2. Find: i) The value of k ii) The sum of the first 9 terms of the AP 6) The 5th and 9th terms of an AP are 37 and 65 respectively. Find: i) The first term and the common difference of the AP. ii) The sum of the first 15th terms. 7) Given that the sum of the first 4 terms is 34 and the sum of the next 4 terms is 82. Find: i) The first term and the common difference of the AP. ii) The sum of the first 10th terms. 8) The 5th term of an AP is 10 and the sum of the first 10 terms of the AP is 115. Find: i) The first term and the common difference of the AP. ii) The sum of the first 20th terms. iii) The sum of the first 10th terms after the 10 terms. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 3.2.3 ARITHMETIC MEAN If a, b, c are three consecutive terms of an AP, then : b–a=c–b b is the arithmetic 2b = a + c mean of a and c b=a+c 2 Example 4: 1 a) Determine whether the following sequences is an AP: 8, 12 , 17, 2 …. b) 2m, 4m + 1, 14 are the three consecutive terms of an AP. Find the value of m. c) Find arithmetic mean for 3 and 25. d) Find 3 arithmetic means for 3 and 11. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 e) Find 4 arithmetic means for 3 and 23. 3.2.4 SOLVING PROBLEMS RELATED TO AP Example 5: a) Ismail has RM 1760 in his bank account in January. Starting from February, he withdraws RM 135 monthly from his bank account. Find his bank balance at the end of December in the same year. b) Mr Siva rents a house with a monthly rental of RM420. in the agreement, it is stated that the monthly rental will increase by the same amount each year. In the 8th year, Mr Siva has to pay a monthly rental of RM560. Calculate : i. The increase in the yearly rental ii. The total amount of house rental Mr Siva has to pay from the 6th year to the 12th year. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 3.3 GEOMETRIC PROGRESSION (GP) 3.3.1 THE nth TERM OF A GP A GP is a number sequence where each term after the first term is obtained by multiplying the preceding term by a constant known as the common ratio. The common ratio cannot take the values 0 or 1 but can be positive or negative. The common ratio,r is given by : The nth term, Tn of an GP is given by : r = Tn+1 Tn = ar n – 1 Tn Where: a = first term Where: T n+1 = (n+1) th term r= common ratio Tn = n th term A GP can be written as a, ar, ar 2, ar 3, ..... Example 6: a) Calculate the 7th and 15th terms of the GP: 4, 12, 36, .... Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 3 1 1 b) Calculate the 6th and 10th terms of the GP: , , ,.... 2 2 6 1 c) What is the number of terms in the GP 81, 27, 9, …., ? 27 1 1 1 d) Given that the nth term of the GP , , , ….. is 128. Find the 32 8 2 value of n. e) Find the nth term of the GP -8, -4, -2, …. in terms of n. Hence, find the 6th term of the GP. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 f) The 2nd and 5th terms of a GP are 1 and -8 respectively. Find : i) The first term and the common ratio ii) The 7th term Exercise 3: 1) Find the nth and 8th term of the GP : a. -3, 9, -27, 81, ...... b. 16, -8, 4, 2, …….. 2) For the following GP, calculate the value for the 10th and 15th terms : a. 1,3,5,7,......... b. 2,4,8,16,........ 1 3) What is the number of terms in the GP 256, 64, 16, …., ? 64 243 4) Given that the nth term of the GP 64, 48, 36 ….. is . Find the 16 value of n. 9 3 5) Find the nth term of the GP , , 1, …. in terms of n. Hence, find the 4 2 8th term of the GP. 6) Given that x + 10, x + 1 and x are the first 3 terms of a GP. Find the value of x and the common ratio. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 7) For a GP, the 3rd term exceeds the first term by 9 and the sum of the 2nd and 3rd terms is 18. Find : i) The first term and the common ratio. ii) The 6th term 3.3.2 SUM OF THE FIRST n TERMS OF A GP The sum of the first n terms, Sn of a GP is given by : Sn = a ( 1 – r n ) 1–r Where: a = first term r = common ratio Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 Example 7: a) Calculate the sum of the first 7 terms of the GP 4, 10, 25, .... 1 b) Calculate the sum of the first 5 terms of the GP 5, 1, , .... 5 3 3 3 c) Calculate the sum of the GP , , , ...., 6. 8 4 2 d) The sum of the first n terms of the GP is 189. Given that the first term and the 2nd term are 3 and 6 respectively. Find the value of n. e) Given a GP 27, 9, 3,…., find the sum from the 4th term to the 9th term of the GP. Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 f) The first three terms of a GP are k + 1, k - 3 and k - 6. Find : i) The value of k ii) The sum of the first 6 terms of the GP Exercise 4: 1) Find the sum of the first 6 terms of the GP : a. -2, -1, -1/2 b. 2, -6, 18 2) The sum of the first n terms of the GP 27, 21, 15, …. is – 72. Find the value of n. 25 3) The 4th and 7th terms of a GP are 25 and respectively. Find the 8 sum of the first 3 terms of the GP. 4) Given a GP 10, 8, 6.4,…., find the sum from the 3rd term to the 7th term of the GP. 1031 5) The sum of the first 5 terms of a GP is and the common ratio is 125 2 . Find : 5 i) The first term ii) The sum of the first 5 terms 201 6) For a GP, the 2nd and 5th terms are -6 and . Find : 4 i) The common ratio Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 ii) The sum of the first 5 terms 7) The first three terms of a GP are h + 2, h - 4 and h - 7. Find : i) The value of h ii) The sum of the first 8 terms of the GP 3.3.3 SUM TO INFINITY OF A GP For the case of -1 < r < 1, the sum of the first n terms of a GP is given by : S∞ = a S∞ is read as 1–r sum to infinity Example 8: a) Find the sum to infinity of each of the following GP : 5 5 a. 5, , , .... 3 9 1 4 1 1 b. 1, , , , , .... 2 3 4 8 1 1 c. 4, -2, 1, − , , 2 4 Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 b) The sum to infinity of a GP with a first term of 12 is 9.6. Find the common ratio of the GP. 3 c) The sum to infinity of a GP with a common ratio of is 32. Find the 4 first term of the GP. 16 d) For a GP, the first term is 18 and the 4th term is . Find : 3 i. The common ratio ii. The sum to infinity of the GP Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 e) For a GP, the first term exceeds the 2nd term by 8 and the sum to infinity is 18. Find the common ratio of the GP 3.3.4 GEOMETRIC MEAN If a, b and c are three consecutive terms of a geometric progression then : b=c a=b b is the b 2 = ac geometric mean b = √ ac of a and c Example 9: a) Determine whether the following sequences is a GP: 3, 9, 27, 81, …. b) Given that m - 8, m - 4, m + 8 are the three consecutive terms of a Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 GP. Find the value of m. c) Find 3 geometric mean for 3 and 23 d) Find 3 geometric mean for 10 and 100 000 e) Find 5 geometric mean for 4 and 2916 Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 3.3.4 SOLVING PROBLEMS RELATED TO GP Example 10: a) A particle moves along a straight line from point P. it covers a distance of 64 cm in the 1st second, 48 cm in the 2nd second, 36 cm in the 3rd second and so on. Find its distance from point P 9 seconds later. b) The price of a house in a certain residential area is RM 220 000. Its price increases 5% each year. Calculate the minimum number of years needed for the price to be more than RM 400 000 for the first time. c) Ibrahim saves 5 cent on the first day, 10 cent on the second day, 20 cent on the third day and so on such that the amount of money he saves on each day is twice that of the previous day. Calculate the Anisah Ahmad JMSK Jun 2011
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BA301 : ENGINEERING MATHEMATHICS 3 minimum number of days needed for the total amount of money to be more than RM 1000 for the first time. Anisah Ahmad JMSK Jun 2011
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