Number Series and Fibonacci Number Series PowerPoint Presentation

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NUMBER SERIES

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INTRODUCTION: A sequence of numbers is known as number series. There are three types of questions coming. Such as: Find the next term in single line series. (ii) Find the odd term in single line series. (iii) Find out the value in two line series.

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TYPES OF SERIES EVEN SERIES: The series in which all the numbers are even numbers followed by a sequence ,is known as even series. e.g. 2,4,6,8,10,12,14,16,18,20etc

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ODD SERIES: The series in which all the numbers are odd numbers followed by a sequence, is known as odd series. e.g.1,3,5,7,9,11,13,15,17,19,21etc.

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PRIME SERIES: The series in which all the numbers are prime numbers followed by a sequence, is known as prime series. e.g.2,3,5,7,11,13,17,19,23,29,31 etc.

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SQUARE SERIES: The series in which all the numbers are squares of natural numbers followed by a sequence is known as square series. e.g.1,4,9,16,25,36,49,64,81,100,121etc.

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CUBE SERIES: The series in which all the numbers are the cube of natural numbers followed by a sequence, is known as cube series. e.g. 1,8,27,64,125,216,343,512,729 etc. 1,4,7,10,13,16,19,22,25,28,31,34etc.

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ARITHMETIC SERIES: The series in which the difference between each successive terms is constant, is known as arithmetic series. e.g.84,80,76,72,68,64,60,56,52,48,44 etc.

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Two tier Arithmetic series: The series in which the difference is constant after two steps, is known as Two tier Arithmetic series. e.g.2,6,12,20,30,42,56,72,90,120 etc.

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In this series, after first step the difference are 4,6,8,10,12,14,16,18,20. And, after second step the differences are 2,2,2,2,2,2etc in all the cases.

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Three tier Arithmetic Series: The series in which the difference is constant after three steps, is known as Three-tier Arithmetic series. e.g.336,210,120,60,24,6,0etc.

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In this case after first step, the differences are 126,90,60,36,18,6. After second step, the differences of these differences are 36,30,24,18,12. After third step ,the differences are 6,6,6,6….

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GEOMETRIC SERIES: The series in which the ratio between each successive terms is constant, is known as geometric series. e.g.1,3,9,27,81,243,729 etc. 1024,512,256,128,64,32 etc. N:B: In this case we apply multiplication or division.

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A.P-G.P SERIES: The series in which we get the next term first by adding/subtracting and then with that result by multiplying/dividing, is known as A.P-G.P series. e.g. 2,5,11,23,47,95,191 etc. 1024,511,254.5,126.25 etc.

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G.P-A.P SERIES: The series in which we can get the next term first by multiplying/dividing and then by adding/subtracting, is known as G.P-A.P series. e.g.3,7,15,31,63,127,255,511,1023 etc. N:B:A.P.-G.P. is also G.P.-A.P. and vice versa.

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TWIN SERIES: The series there are two sequences with in one is known as twin series. e.g.2,4,7,9,12,14,17,20,22,25 etc. 2,3,6,9,18,27,54,81,162 etc.

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METHOD TO GET THE SERIES A) If the rate of increase/decrease is slow in a series, there must be addition or subtraction. So the series is an Arithmetic series.

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(B) If the rate of increase/decrease is high in a series, there may be addition of squares or cubes or multiplication/division, and also multiplication then addition. So first take the difference between some of the successive terms. If that are squares/cubes, the series is an A.P. other wise is G.P.or G.P-A.P.

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(A)2,6,12,20,30,42,56,72,…. (B)2,3,11,38,102,227,443,…. (C)3,4,8,17,33,58,94,143,…. (D)2,2,4,12,48,240,1440,….. (E) 3,4,10,33,136,685,…..

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ANSWERS (A)90 (B)786 (C)207 (D)10080 (E)4116

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SOLUTION: (A) The series increase slowly. So it is from A.P. The differences are 4,6,8,10 etc, so the next term is 90. (B) The series increases high, but the differences are cubes. So it is from A.P.& the next term is 886. (C) The series increases high, but the differences are squares. So it is from A.P.& the next term is 207. (D) The series increases high, but the differences are not squares or cubes & there is ratio between each successive terms. So it is from G.P. & the next term is 10080. (E) The series increases high, but the differences are not squares or cubes & there is multiplication and addition. So it is from G.P.-A.P & the next term is 4116.

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TWO LINE SERIES: Introduction: In this type of series one complete series must be given while the other is incomplete. By applying the sequence available in the first line you have to determine the required number of the second line series.

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EXAMPLE: 4, 5, 13, 40, 104 229 64,(a), (b), (c), (d) (e) Find the value of (e) a)269 (b)279 (c)289 (d)299 (c)289

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SOLUTION: The rate of increase is high, but the differences are the cubes. So the series is an A.P. The value of (e) is 289.

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METHOD TO GET THE REQUIRED TERM If the rate of increase is slow. If the rate of increase is high.

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If the rate of increase/decrease is slow, it is from Arithmetic series. Take the difference between the first term of the two lines. If the first term of second line is higher than the first term of first line add the difference with the required number of the first line. If the first term of second line is lower than the first term of first line subtract the difference from the required number of the first line.

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EXAMPLE: 128, 124, 118, 110, 100 100, (A), (B), (C), (D), Find the value of (D) (a) 70 (b) 72 (c) 74 (d) 76 (b) 72

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SOLUTION: The rate of decrease is slow. So that the series is an A.P. The first number of second line is less than the corresponding value of first line. So the value of (d) is 72.

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If the rate of increase is high, there may be addition of squares/cubes or multiplication or multiplication with addition. So for the first step take the difference between some of the successive terms & mark whether that are the squares/cubes or not. If that are the squares/cubes, then there is the addition of squares/cubes. Other wise there may be only multiplication or multiplication and then addition,

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EXAMPLE 4, 4, 8, 24, 96, 480 6,(a),(b),(c),(d),(e) Find the value of (e) (a)720 (b)725 (c)750 (d)760 (a) 720

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SOLUTION: The rate of increase is high and the differences are not squares or cubes and also there is ratio between the terms. So the series is from G.P. By taking the ratio 4 parts=480 6 parts=720

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QUESTIONS & ANSWERS

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QUESTION 1: (A) 4, 5, 13, 40, 104, 229 64, (a), (b), (c), (d), (e) Find the value of (e) (a)279 (b)289 (c)278 (d)299 (b) 289

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SOLUTION: The rate of increase is high, but the difference between the successive terms are the cubes. So the series is from A.P. The difference between the first term of two lines is 60 & the second line number is larger . So the value of (e) is 60 more than the corresponding first line i.e.289

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QUESTION 2: 3, 3, 6, 18, 72, 360 5, (a), (b), (c), (d), (e) Find the value of (e) (a)500 (b)600 (c)625 (d)650 (b) 600

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SOLUTION: The rate of increase is high & the differences are not squares/cubes but all the terms are multiples of the first term. So by taking the ratio 3 parts=360 So 5 parts=600.

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QUESTION 3: 3, 3, 9, 45, 315 5, (A), (B), (C), (D) Find the value of (D) (a)500 (b)520 (c)525 (d)550 (c) 525

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SOLUTION

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QUESTION 4 4, 14, 36, 114, 460 2, (A), (B), (C), (D),(E) Find the value of (E) (a)2060 (b)2062 (c)2064 (d)2068 (b) 2062

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QUESTION 5: 5, 6, 11, 28, 71, 160 2, 3, (A), (B), (C), (D), (E) Find the value of (D) (a)157 (b)156 (c)154 (d)155 (a) 157

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6) 1296,864, 576, 384,256 1080, (A), (B), (C), (D),(E) Find the value of (D). 7) 7, 13, 78, 83, 415 3, (A), (B), (C), (D),(E) Find the value of (C).

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8) 3240, 540,108, 27, 9 3720, (A), (B), (C),(D),(E) Find the value of (D). 9) 27, 44, 71, 108,155 36, (A), (B), (C), (D), (E) Find the value of (C).

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10) 108, 52, 24, 10, 3 64, (A), (B), (C), (D),(E) Find the value of (D). 11) 3, 2, 10, 4, 28 2, (A) , (B), (C), (D),(E) Find the value of (C).

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12) 2, 5, 17.5, 43.75, 153.125 1, (A), (B), (C), (D),(E) Find the value of (D) 13) 3, 6, 24, 72, 144, 576 1, (A), (B), (C), (D), (E) Find the value of (E).

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14) 575, 552, 533, 518, 507 220, (A), (B), (C), (D), (E) Find the value of (E).

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The Fibonacci numbers are Nature's numbering system. The Fibonacci Sequence is the series of numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... Longer list: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, ... Fibonacci Number

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Thank You