Arithmetic sequence formula calculator

arithmetic sequence tutorial and algebra 2 arithmetic sequences worksheet
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Sequences MATH 1510 Lili Shen Sequences Definition and Summation + // A sequence is a function a :N R. The terms of the Notation sequence are the function values, usually denoted by Arithmetic Sequences a = a(n) n + for all n2N . So the terms of the sequence are written as a ;a ;:::;a ;:::; 1 2 n where a is the first term and, in general, a is the nth term. 1 nTerms of a sequence MATH 1510 Lili Shen Sequences Example and Summation Notation Find the first five terms and the 100th term of the sequence Arithmetic defined by each formula: Sequences (1) a = 2n 1. n 2 (2) c = n 1. n n (3) t = . n n+ 1 n (1) (4) r = . n n 2Terms of a sequence MATH 1510 Lili Shen Sequences Solution. and Summation (1) a = 1, a = 3, a = 5, a = 7, a = 9, a = 199. Notation 1 2 3 4 5 100 Arithmetic (2) c = 0, c = 3, c = 8, c = 15, c = 24, c = 9999. 1 2 3 4 5 100 Sequences 1 2 3 4 5 100 (3) t = , t = , t = , t = , t = , t = . 1 2 3 4 5 100 2 3 4 5 6 101 1 1 1 1 1 (4) r = , r = , r = , r = , r = , 5 1 2 3 4 2 4 8 16 32 1 r = . 100 100 2Terms of a sequence MATH 1510 Lili Shen Sequences and Summation Notation Example Arithmetic Sequences Find the nth term of a sequence whose first several terms are given: 1 3 5 7 (1) , , , ,::: . 2 4 6 8 (2)2, 4,8, 16,32,::: .Terms of a sequence MATH 1510 Lili Shen Sequences and Summation Notation Solution. Arithmetic Sequences 2n 1 (1) a = . n 2n n n (2) a = (1) 2 . nRecursively defined sequences MATH 1510 Lili Shen Sequences and Summation Notation Some sequences do not have simple defining formulas like Arithmetic Sequences those of the preceding example. The nth term of a sequence may depend on some or all of the terms preceding it. A sequence defined in this way is called recursive.The Fibonacci sequence MATH 1510 Lili Shen Sequences and Summation Notation Example (The Fibonacci sequence) Arithmetic Find the first 11 terms of the sequence defined recursively Sequences by F = 1, F = 1 and 1 2 F = F +F n n1 n2 for all n 3.The Fibonacci sequence MATH 1510 Lili Shen Sequences and Summation Notation Arithmetic Solution. Sequences 1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89;::::The Fibonacci sequence and the golden ratio MATH 1510 Lili Shen Sequences and Summation Notation The limit p Arithmetic Sequences F 1+ 5 n+1 ' = lim =  1:618 n1 F 2 n is known as the golden ratio, which is believed as the key to creating aesthetically pleasing art by artists and architects.Golden ratio MATH 1510 Lili Shen The Parthenon in Greece Sequences and Summation Notation Arithmetic SequencesGolden ratio MATH 1510 Lili Shen Sequences and Summation Notation Arithmetic SequencesThe partial sum of a sequence MATH 1510 Lili Shen Definition Sequences and For a sequence Summation Notation a ;a ;:::;a ;:::; 1 2 n Arithmetic Sequences the nth partial sum is defined as S = a +a ++a n 1 2 n + for all n2N . The sequence S ;S ;:::;S ;::: 1 2 n is called the sequence of partial sums.The partial sum of a sequence MATH 1510 Lili Shen Sequences and Summation Notation Arithmetic Example Sequences Find the first four partial sums and the nth partial sum of the 1 sequence given by a = . n n 2The partial sum of a sequence MATH 1510 Solution. Lili Shen The terms of the sequence are Sequences and Summation 1 1 1 1 1 Notation ; ; ; ;:::; ;:::: n 2 4 8 16 2 Arithmetic Sequences Thus 1 S = ; 1 2 1 1 3 S = + = ; 2 2 4 4 1 1 1 7 S = + + = ; 3 2 4 8 8 1 1 1 1 15 S = + + + = : 4 2 4 8 16 16The partial sum of a sequence MATH 1510 Lili Shen Sequences and Summation Notation Arithmetic In general, the nth partial sum is Sequences n 2 1 1 S = = 1 : n n n 2 2Sigma notation MATH 1510 Lili Shen Given a sequence Sequences and Summation Notation a ;a ;:::;a ;:::; 1 2 n Arithmetic Sequences we can write the sum of the first n terms using summation notation, or sigma notation as n X a = a +a ++a : k 1 2 n k=1 The left side of this expression reads as “The sum of a k from k = 1 to k = n.”Sigma notation MATH 1510 Lili Shen Sequences and Summation Notation The letter k is called the index of summation, or the Arithmetic summation variable. Sequences The idea is to replace k in the expression after the sigma by the integers 1; 2; 3;:::;n, and add the resulting expressions, arriving at the right-hand side of the equation.Sigma notation MATH 1510 Lili Shen Example Sequences Find each sum: and Summation 5 X Notation 2 (1) k . Arithmetic Sequences k=1 5 X 1 (2) . j j=3 10 X (3) k. k=5 6 X (4) 2. i=1Sigma notation MATH 1510 Lili Shen Solution. Sequences 5 X and 2 2 2 2 2 2 Summation (1) k = 1 + 2 + 3 + 4 + 5 = 55. Notation k=1 Arithmetic 5 Sequences X 1 1 1 1 47 (2) = + + = . j 3 4 5 60 j=3 10 X (3) k = 5+ 6+ 7+ 8+ 9+ 10 = 45. k=5 6 X (4) 2 = 2+ 2+ 2+ 2+ 2+ 2 = 12. i=1Properties of sums MATH 1510 Lili Shen Proposition Sequences and Summation Let a ;a ;a ;a ;::: and b ;b ;b ;b ;::: be sequences. 1 2 3 4 1 2 3 4 Notation + Thenforalln2N andc2R,thefollowingpropertieshold: Arithmetic Sequences n n n X X X (1) (a +b ) = a + b . k k k k k=1 k=1 k=1 n n n X X X (2) (a b ) = a b . k k k k k=1 k=1 k=1 n n   X X (3) ca = c a . k k k=1 k=1

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