Mathematics prof
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Mathematics is the study of quantity, structure, space, and change. Mathematicians seek patterns and establish truth through rigorous logic and deduction from chosen axioms and definitions. The document then provides examples of different types of sequences including geometric sequences defined by multiplying a number each time, arithmetic sequences defined by adding a number each time, and special sequences like triangular and square numbers generated from patterns. It also discusses how to identify if a sequence is arithmetic, geometric, or harmonic based on relationships between terms.
Mathematics prof
- 1. • Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.
- 2. A list of numbers that follow a certain pattern or sequence
- 4. • Geometric Sequences • A Geometric Sequence is made by multiplying by some value each time. • Example: • 2, 4, 8, 16, 32, 64, 128, 256, ... • This sequence has a factor of 2 between each number. • The pattern is continued by multiplying by 2 each time. • Arithmetic Sequences • An Arithmetic Sequence is made by adding some value each time. • Example: • 1, 4, 7, 10, 13, 16, 19, 22, 25, ... • This sequence has a difference of 3 between each number. • The pattern is continued by adding 3 to the last number each time.
- 5. Special Sequences Triangular Numbers 1, 3, 6, 10, 15, 21, 28, 36, 45, ... This Triangular Number Sequence is generated from a pattern of dots which form a triangle. By adding another row of dots and counting all the dots we can find the next number of the sequence: Square Numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, ... The next number is made by squaring where it is in the pattern. The second number is 2 squared (2^2 or 2×2) The seventh number is 7 squared (7^2 or 7×7) etc
- 7. Identifying whether the sequence is A.P, G.P, H.P If, a–b a A.P b–c a a–b a G.P b–c b a–b a H.P b–c c Arithmetic Mean A= a+ b 2 G2 = AH Geometric Mean G = √ab A>G>H Harmonic Mean H= 2ab a+b
- 8. Some tips : If first common difference is in A.P take the General Term as ‘ax2 + bx +c’ and determine a, b, c by solving for known values.
- 9. If the sum of first n terms of an A.P is cn2 , then the sum of squares of these n terms is (B) n(4n2 + 1)c2 3 (C) n(4n2 – 1)c2 (D) n(4n2 + 1)c2 3 6 Solution : Sn = cn2 *Shortcut Method : Tn = Sn – Sn-1 = cn2 – c(n-1)2 = c(2n – 1) Put n = 1 in the Tn2 = c2(2n – 1)2 Sn = Σ Tn question
- 10. Above pattern is of tables at a function around which a particular number of people must sit. In order to work out how many people can sit around any arrangement of tables you must use a number of different formats. The first of these is to draw the pattern to the sequence in the pattern you need.
- 13. Mathematical Model #seats = slope x #of tables + y-intercept or
- 14. Slope is the rise over the run = --------- Y-intercept is the point where the line crosses the Y axis. =
- 15. #tables #seats 1 4 2 6 3 8 4 10 5 12 6 14
- 17. 2, 4, 6, 8, 10, ____
- 18. Apply patterns • Use the shapes to addition… for help One heart 1 + 1 = ____ plus one heart Five smiles plus three 5 + 3 = ____ smiles Seven 7 + 8 = ____ stars plus eight stars
- 19. Apply patterns to • Use the shapes subtraction… for help Start Finish 3 – 2 = ____ 5 – 3 = ____ 12 – 9 = ____
- 20. (AGRAWAL, KUMAR, & SHAKIR) (Haggert) (Jayakumar) (www.everythingmaths.co.za) (KUMAR & SHAKIR) (learning) Bibliography AGRAWAL, A., KUMAR, A., & SHAKIR, M. B. (n.d.). MATHEMATICS. Haggert, J. (n.d.). Mathematics. Jayakumar, J. (n.d.). The Major Milestone Of Mathematics. KUMAR, A., & SHAKIR, M. B. (n.d.). Mathematics. learning, T. e. (n.d.). Introduction to pattern. www.everythingmaths.co.za. (n.d.). Everything Maths.