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Sequence and Series in Discrete Structure
The presentation discusses sequences and their applications across various fields such as physics, biology, and computer science. It explains different types of sequences, including arithmetic, geometric, and harmonic progressions, along with their means and examples. The advantages of a structured approach to teaching sequences in a two-year curriculum are also highlighted.
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Introduction to the importance of sequences and series in various fields like Physics, Economics, etc.
Definition of sequences; illustrations of simple numerical sequences.
Explanation of sequence as a basic programming algorithm and its relevance in microprocessor operations.
Introduction of necessary mathematical tools for sequences: Arithmetic Progression, Means, and Progressions.
Definition and explanation of Arithmetic Progression; includes examples on finding terms and Arithmetic Mean.
Definition and explanation of Geometric Progression; examples provided for terms and geometric mean.
Definition and explanation of Harmonic Progression; includes examples for finding terms and Harmonic Mean.
Describes the uses of sequences in representing ordered lists and discusses the advantages of structured sequences in education.
A closing thank you slide.
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Sequence and Series in Discrete Structure
- 1. National College ofBusiness Administration & Economics
- 2. Group Members -40 Muhammad Ahsan ShahidZafar Sarmad Shuja Wamiq Ali Zain Abid 01 02 03 04 05
- 3. INSERT THE TITLE OFYOUR PRESENTATION HERE DS PPT
- 4. The sequence andseries have important applications in Physics, Geography, Biology, Economics, Psychology, Engineering, Chemistry, Computer science and Finance. Introduction
- 5. Sequence A sequence isan 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, …
- 6. 𝟐𝟓, 𝟐𝟏, 𝟏𝟕,𝟏𝟑, 𝟗, … . Example:
- 7. Sequence in ComputerScience In Programming, sequence is the basic algorithm: A set of logical steps carried out in order that commands are executed by a computer, allows us to carry out tasks that have multiple steps. Microprocessor also follow a sequence which is Fetch, Decode, and then Execute.
- 8. Necessary Tools ForMicroprocessor Arithmetic Progression AP 01 Arithmetic Mean AM 02 Geometric Progression GP 03 Harmonic Progression05 Geometric Mean GM 04 Harmonic Mean06 HP HM
- 9. Arithmetic Progression An ArithmeticProgression is a sequence in which every term is obtained by adding some fix number to the preceding term. The fix number to preceding term. The fix number is the common difference of two consecutive terms denoted by d.
- 10. Find the 10thterm if 𝑎3 = 8 and 𝑎7 = 16 Example:
- 11. Arithmetic Mean Let aand b be two numbers, than a number A is set to be A.M between a and b if a, A, b are in A.P Thus A - a = b – A 2A =a +b A = a+b/2
- 12. Find A.M between 2𝑥+ 1, 4𝑥 + 3 Example:
- 13. Geometric Progression A sequencein which every term is obtained by multiplying or dividing a definite nu mber with the preceding number is known as a geometric sequence. 𝑟 = 𝑎 𝑛 𝑎 𝑛−1 𝑛 > 1 , ∀𝑛 ∈ 𝑁, 𝑎 𝑛−1 ≠ 0
- 14. Find 𝑎 𝑛if 𝑎4 = 8 27 and 𝑎7 − 64 729 of a G.P. Example:
- 15. Geometric Mean Let aand b two numbers. A number G is said to be geometric mean between two numbers a and b if a, G, b are in G.P,so 𝐺 𝑎 = 𝑏 𝐺 𝐺2 = 𝑎𝑏 𝐺 = ±𝑎𝑏
- 16. Find G.M between 2iand -4i Example:
- 17. Harmonic Progression If thereciprocal of the terms of the sequence form an A.P, then sequence is called harmonic.
- 18. Find the 8thterm of H.P: 1 2 , 1 5 , 1 8 , … Example:
- 19. Harmonic Mean Let aand b be two numbers, then a number H is called Harmonic mean between t wo numbers a and b if a,H,b are in H.P
- 20.
- 21. Uses of Sequence A sequence is a discrete structure used to represent an ordered list. A sequence is a function from a subset of the set of integers (usually either the set {0,1,2.. ..} Or {1,2, 3,...}To a set S. We use the notation an to denote the image of the integer n..... Notation to represent sequence is {an}
- 22. Advantages: The common 2-yearsequence works well for many disciplines. Topics can be introduced "just-in-time" for many disciplines. Since all students take the same sequence, advising is relatively easy.
- 23.