Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Abuse of notation wikipedia , lookup
Hyperreal number wikipedia , lookup
Large numbers wikipedia , lookup
Collatz conjecture wikipedia , lookup
Series (mathematics) wikipedia , lookup
Proofs of Fermat's little theorem wikipedia , lookup
Elementary mathematics wikipedia , lookup
10/1/2013 Definition: Sequences & Sigma Notation Trigonometry Radnor High School A sequence is function whose domain is the set of natural (counting) numbers. The outputs of a sequence are called terms. For convenience, convenience we often refer to the terms of the sequence as the sequence involved. Normally, sequence is written as a list of numbers written in a definite order: a1, a2, a3,…, an,… The number a1 is called the first term, a2 is called the second term and so on. 2 A sequence can be represented as a list of numbers, or defined by a closed form rule for the nth term, or defined recursively (first term(s) are given and rule to find the next term). Example 1: Write the next three terms of the following sequences: a ) 4, 8, 14, 22, 32, 44, ... For example: a) 1, 1 5, 5 7, 7 3, 3 4, 4 8, 8 10, 10 …. b) 3, 7, 11, 15, 19, … b) 3, 6, 12, 24, 48, .... c) an = 3n +1 d) b1 = 5, bn+1 = 3bn + 4 Note: Certain sequences have patterns between consecutive terms. Knowing these patterns, we can write a rule for the general nth term of the sequence. patterns frequently q y occur are: Some of the p c) 1 4 7 10 , , , ,... 3 9 27 81 Example 2: Write a closed form rule for the nth term of the sequence below: 2, 5, 12, 19, 26, 33, ... a) The first differences are constant (add/subtract the same number to get the next term) linear function b) The second differences are constant quadratic function (use regression on the calculator to find the rule) c) Multiply the same number to get the next term. 1 10/1/2013 Example 3: Example 4: Write a closed form rule for the nth term of the sequence below: 6,9,16, 27, 42, 61,... Example 5: Write a closed form rule for the nth term of the sequence below: 3 5 7 9 11 , , , , ,... 5 9 13 17 21 Sigma Notation Write the first five terms of each sequence below: k: ending number for n a) an n 2 k b) bn 1 n n n3 : add the terms n i c) cn n 1 ! Example 6: Evaluate each sigma notation by writing out the terms: Evaluate each sigma notation by writing out the terms: a) k b) j 1 6 k 2 j 5 n 2 3 an: terms to add Example 7: n 3 b) n i: starting number for n 7 a) a 2 3 4 3n 2 n 0 2