Slope of a Line - City University of New York

Chapter 12 Discrete Algebra 12.1 Sequences and Sums Sequence A sequence is an ordered list of numbers, suc h as 1, 3, 5, 7, 9, 1, -3, 5, -7, 9, -11, 13, 1, 0, 1, 0, 1, 0, 2, 1, 2/ 3, 3/ 4, 4/ 5, 5/ 6, The numbers in the sequences are called terms. 1, -3, 5, -7, 9, -11 First term Second term Third term

Fourth term Fifth term A finite sequence has a last term. ex. 4, 1, -2, -5, -8 An infinite sequence continues without stopping. ex. 4, 1, -2, -5, -8, Sixth term Example Example: Write the first six terms of the sequence for an 2n 3 Solution: a 2(1) 3 5 1 a 2(2) 3 7 2 a 2(3) 3 9 3 a 2(4) 3 11 4 a 2(5) 3 13

5 Example Example: Write the first three terms, the 35th term, and the 294th term of the sequence Solution: ( 1) n . n2 ( 1)1 a 1 1 1 2 3 ( 1) 2 a 1 2 22 4

( 1)3 a 1 3 5 32 ( 1)35 a 1 35 35 2 37 ( 1) 294 a 1 294 294 2 296 Summation Notation The summation notation can be used to write a series. Mathematicians

often use the Greek letter sigma, written as , to abbreviate a sum. Summation notation m Ck means c1 c2 c3 ... cm k 1 The letter k is called the summation index. Any letter may be used for the summation index. Series When the terms of a sequence are added, the resulting expression is a series. A series can be infinite or finite. 5 Finite series: 3i 3 6 9 12 15 i1 Infinite series:

3i 3 6 9 12 15 ... i1 Kth Partial Sum The sum of the first k terms of the sequence is c alled the kth partial sum of the sequenc e. The kth partial sum of a is n k an a1 a2 a3 ... ak n1 Example: Find the sum of the series. 5 a) [( 1) k 2k ] k 1

1 5 5 9 9 29 7 b) (3 j 4) j 3 5 8 11 14 55 Properties of Sums r 1. can c a for any number c . n n1 n 1 r r r

r n1 n1 n1 r r r 2. (an bn ) an bn 3. (an bn ) an bn n1 n1 n1

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