* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Number Sequence
Survey
Document related concepts
Positional notation wikipedia , lookup
Law of large numbers wikipedia , lookup
Location arithmetic wikipedia , lookup
History of logarithms wikipedia , lookup
Georg Cantor's first set theory article wikipedia , lookup
Abuse of notation wikipedia , lookup
Functional decomposition wikipedia , lookup
Karhunen–Loève theorem wikipedia , lookup
Elementary algebra wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Factorization wikipedia , lookup
Non-standard analysis wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Collatz conjecture wikipedia , lookup
Large numbers wikipedia , lookup
Hyperreal number wikipedia , lookup
Transcript
Section 1-2 An Application of Inductive Reasoning: Number Patterns Slide 1-2-1 AN APPLICATION OF INDUCTIVE REASONING: NUMBER PATTERNS • • • • Number Sequences Successive Differences Number Patterns and Sum Formulas Figurate Numbers Slide 1-2-2 NUMBER SEQUENCES Number Sequence A list of numbers having a first number, a second number, and so on, called the terms of the sequence. Arithmetic Sequence A sequence that has a common difference between successive terms. Geometric Sequence A sequence that has a common ratio between successive terms. Slide 1-2-3 SUCCESSIVE DIFFERENCES Process to determine the next term of a sequence using subtraction to find a common difference. Slide 1-2-4 EXAMPLE: SUCCESSIVE DIFFERENCES Use the method of successive differences to find the next number in the sequence. 14, 22, 32, 44,... 14 22 8 32 10 2 14 12 2 58 44 2 Find differences Find differences Build up to next term: 58 Slide 1-2-5 NUMBER PATTERNS AND SUM FORMULAS Sum of the First n Odd Counting Numbers If n is any counting number, then 1 3 5 (2n 1) n . 2 Special Sum Formulas For any counting number n, (1 2 3 n)2 13 23 n3 n(n 1) and 1 2 3 n . 2 Slide 1-2-6 EXAMPLE: SUM FORMULA Use a sum formula to find the sum 1 2 3 48. Solution Use the formula 1 2 3 with n = 48: n(n 1) n 2 48(48 1) 1176. 2 Slide 1-2-7 FIGURATE NUMBERS Slide 1-2-8 FORMULAS FOR TRIANGULAR, SQUARE, AND PENTAGONAL NUMBERS For any natural number n, n(n 1) the nth triangular number is given by Tn , 2 the nth square number is given by Sn n2 , and n(3n 1) the nth pentagonal number is given by Pn . 2 Slide 1-2-9 EXAMPLE: FIGURATE NUMBERS Use a formula to find the sixth pentagonal number Solution n(3n 1) Use the formula Pn 2 with n = 6: 6[6(3) 1] P6 51. 2 Slide 1-2-10