PATTERNS ( Mathematics in the modern world)-1.docx
- 1. PATTERNS Nature of Mathematics Mathematics can be described as the study of abstract structures and patterns that can be expressed in symbolic language. It is a systematic approach to understanding the relationships between numbers, quantities, and shapes, and how they can be used to solve problems and make predictions. Mathematics is a universal language that transcends cultural and linguistic barriers. It is used in a variety of fields such as physics, engineering, economics, and computer science to model and analyze complex systems. Mathematics is also a discipline that values precision, logical reasoning, and creativity. It is both an art and a science, and its practitioners often strive to find elegant and beautiful solutions to problems. Mathematics plays a fundamental role in our understanding of the natural world and in the development of new technologies that shape our daily lives. Patterns in Nature Patterns in nature and mathematics share intriguing connections, as mathematics is often used to describe, understand, and explain the patterns observed in the natural world. These patterns can be found across different scales, from the microscopic to the cosmic, and they often result from the fundamental laws of physics and the processes of evolution and self-organization. Some common examples of patterns in nature include: 1. Fractals Fractals are complex geometric patterns that exhibit self-similarity at different scales. They can be found in various natural phenomena, such as snowflakes, coastlines, trees, and clouds. https:// t henewcode.c om/assets/ images/ thumbnails/ aloe- polyphylla- 2x.jpg https:// live.staticflickr.com/2675/3914828202_66770e03b6_n.jpg
- 2. 2. Fibonacci Sequence The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones (e.g., 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on). This sequence is often seen in the arrangement of leaves on a stem, the spirals of sunflower seeds, and the growth patterns of certain seashells. https://i.pinimg.com/originals/05/6b/ 77/056b77ab420e5d7624e08c57c2c66b79.png https://i.pinimg.com/originals/22/09/92/220992f678ba38f26d62ce7865de2d4a.jpg 3. Symmetry Symmetry is a fundamental pattern found in many natural objects, such as butterflies' wings, flowers, and crystals. It represents a balance and regularity of form that is aesthetically pleasing and often associated with biological fitness.
- 3. https://pbs.twimg.com/profile_images/1143390737536565248/Z56wfjKC.png https://i.pinimg.com/736x/29/e4/65/29e465477444ab6b1899491010ab9b15.jpg 4. Spiral Patterns Spirals are prevalent in nature and can be observed in phenomena like the shape of galaxies, hurricanes, and the growth patterns of certain plants (e.g., pinecones and nautilus shells). https:// i.pinimg.com/474x/cc/5f/89/ cc5f8973f71102512bc6cf89c2817f86--fern-frond-nature-scenes.jpg
- 4. https://i.pinimg.com/736x/20/1d/2f/201d2fa633df000fdf1262ca8b8ae445--indoor- plants-tag-image.jpg 5. Hexagonal Packing Hexagonal patterns are common in natural structures due to their efficient packing arrangement. For example, honeybee combs are composed of hexagonal cells that maximize storage and structural integrity. https:// live.staticflickr.com/ 193/497929625_5168bef6e2_m.jpg 6. Wave Patterns Wave patterns can be observed in various natural phenomena, such as ocean waves, sand dunes, and the ripples formed when a stone is thrown into a pond. https://i.pinimg.com/originals/33/e1/be/33e1be096ee1f7fc9325b4776deff8dd.jpg
- 5. https://th.bing.com/th/id/OIP.Hwb8objXUbz7_0Z7pjx_mgHaE8?pid=ImgDet&rs=1 7. Camouflage and Mimicry Many animals display patterns that help them blend into their surroundings (camouflage) or imitate other organisms to deter predators or catch prey (mimicry). https:// www.bioedonline.org/ BioEd/cache/file/6F5589C7-F38A-4886-B06622DBDE5A6368.jpg https:// i.pinimg.com/ originals/5d/ 62/1a/ 5d621a645f47c2e961e3e650e722d85e.jpg
- 6. 8. Seasonal Changes The changing patterns of the seasons, driven by the tilt of the Earth's axis, create cyclic variations in weather, plant growth, and animal behavior. https://resize.hswstatic.com/w_830/gif/redefinefourseasons-2.jpg 9. Ecosystem Patterns Ecosystems often exhibit patterns of diversity and interdependence among species, resulting in complex food webs and ecological relationships. https://th.bing.com/th/id/OIP.G0AO_ovuaCd_CZEpxQfVKQAAAA?pid=ImgDet&rs=1
- 7. https:// i.pinimg.com/736x/3b/07/92/3b07923117edb554a485cff3d232b30b.jpg 10. Chaos and Complexity Some natural systems, like weather patterns or the behavior of certain animal populations, exhibit chaotic behavior and complex dynamics, which can lead to intricate patterns.
- 8. https://thumbs.dreamstime.com/b/tangled-group-arching-branches-autumn-leaves- covered-vibrant-green-mosses-which-love-damp-conditions-235114202.jpg https://i1.sndcdn.com/artworks-qLzwvPUEsiwpansZ-fJtVWw-t500x500.jpg These patterns in nature not only inspire aesthetic appreciation but also serve as valuable sources of knowledge for scientists and mathematicians seeking to understand the underlying principles and laws that govern the natural world. Observing and studying these patterns can lead to new insights and discoveries across various scientific disciplines. Sequence and Series Sequence and series are the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas a series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series. In short, a sequence is a list of items/objects which have been arranged in a sequential way. A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.
- 9. Sequence and Series Definition A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a1, a2, a3, a4,……… etc. denote the terms of a sequence, then 1,2,3,4,…..denotes the position of the term. A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence. If a1, a2, a3, a4, ……. is a sequence, then the corresponding series is given by SN = a1+a2+a3 + .. + aN Note: The series is finite or infinite depending if the sequence is finite or infinite. Types of Sequence and Series Some of the most common examples of sequences are: Arithmetic Sequences Geometric Sequences Harmonic Sequences Fibonacci Numbers Arithmetic Sequences A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence. Geometric Sequences A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence. Harmonic Sequences
- 10. A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence. Fibonacci Numbers Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F0 = 0 and F1 = 1 and Fn = Fn- 1 + Fn-2 Sequence and Series Formulas List of some basic formula of arithmetic progression and geometric progression are Arithmetic Progression Geometric Progression Sequence a, a+d, a+2d, ……,a+(n-1)d,…. a, ar, ar2 ,….,ar(n-1) , … Common Difference or Ratio Successive term – Preceding term Common difference = d = a2 – a1 Successive term/Preceding term Common ratio = r = ar(n-1) /ar(n-2) General Term (nth Term) an = a + (n-1)d an = ar(n-1) nth term from the last term an = l – (n-1)d an = l/r(n-1) Sum of first n terms sn = n/2(2a + (n- 1)d) sn = a(1 – rn )/(1 – r) if |r| < 1 sn = a(rn -1)/(r – 1) if |r| > 1
- 11. *Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term Difference Between Sequences and Series Let us find out how a sequence can be differentiated with series. Sequences Series Set of elements that follow a pattern Sum of elements of the sequence Order of elements is important Order of elements is not so important Finite sequence: 1,2,3,4,5 Finite series: 1+2+3+4+5 Infinite sequence: 1,2,3,4, …… Infinite Series: 1+2+3+4+ …… Sequence and Series Examples 1. If 4,7,10,13,16,19,22……is a sequence, Find: 1. Common difference 2. nth term 3. 21st term Solution: Given sequence is, 4,7,10,13,16,19,22…… a) The common difference = 7 – 4 = 3 b) The nth term of the arithmetic sequence is denoted by the term an and is given by an = a + (n-1)d, where “a” is the first term and d is the common difference.
- 12. an = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1 c) 21st term as: a21 = 4 + (21-1)3 = 4+60 = 64. 2. Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term. Solution: The common ratio (r) = 4/1 = 4 The preceding term is multiplied by 4 to obtain the next term. The nth term of the geometric sequence is denoted by the term an and is given by an = ar(n-1) where a is the first term and r is the common ratio. Here a = 1, r = 4 and n = 9 So, 9th term is can be calculated as a9 = (1)(4)(9-1) = 48 = 65536. 3.Find the eight element of the geometric sequence for which the first element is 3/2 and the second element is -3. Solution: r = -3 ÷ 3/2 = -2 a = 3/2 , n = 8 a8 = ar(n-1) a8 = (3/2)(-2)(8-1) a8 = (3/2)(-128) a8 = -192 4. Find the 10th term of the arithmetic sequence 2, 4, 6, 8, 10, … In this sequence, the first term is 2, and the common difference is 2. To find the 10th term, we can use the formula: a10 = 2 + (10-1)(2) = 2 + 18
- 13. a10 = 20 5. Find the 8th term of the arithmetic sequence -3, -1, 1, 3, 5… In this sequence, the first term is -3, and the common difference is 2. To find the 8th term, we can use the formula: a8 = -3 + (8-1)(2) = -3 + 14 a8 = 11 6. Find the sum of the first 7th term of this series 7 + 14 + 21 + 28 +... In this sequence, the first term is 7, and the common difference is 7. To find the sum of the first 7 terms, we can use the formula: Sn = n 2 (a + an) S7 = (7/2)(7 + 49) = (7/2)(56) S7 = 196 7. Find the sum of the first 8th term of this series 12 + 8 + 4 + 0 +... In this sequence, the first term is 12, and the common difference is -4. To find the sum of the first 8 terms, we can use the formula: Sn = n 2 (a + an) S8 = (8/2)(12 + -16) = (4)(-4) S8 = -16 So, the sum of the first 8 terms is -16.