1 Section 3.2 Sequences and Summations

2 Sequence Function from a subset of Z (usually the set beginning with 1 or 0) to a set S a n denotes the image of n (n  Z) a n is called a term of the sequence The notation {a n } is used to describe the sequence

3 Example Find the first 5 terms of sequence {a n } where {a n } = 2 * (-3) n + 5 n a 0 = 2 * (-3) = 2 * = 3 a 1 = 2 * (-3) = 2 * (-3) + 5 = -1 a 2 = 2 * (-3) = 2 * = 43 a 3 = 2 * (-3) = 2 * (-27) = 71 a 4 = 2 * (-3) = 2 * = 786

4 Strings Strings are finite sequences of the form: a 1, a 2, a 3, …, a n The number of terms in a string is the length of the string The empty string has 0 terms

5 Special Integer Sequences Find the formula or general rule for constructing the terms of a sequence, given a few initial terms –Look for a pattern in the terms you’re given –Determine how a term can be produced from a preceding term

6 Useful clues for special integer sequences Runs of a value Terms obtained by adding to previous term: –the same amount –an amount that depends on the term’s position in the sequence Terms obtained by multiplying the previous term by some amount Terms obtained by combining previous terms

7 Examples 1,0,1,1,0,0,1,1,1,0,0,0,1,… Both 1 and 0 appear exactly n times, alternating 1,2,2,3,4,4,5,6,6,7,8,8,… The positive integers appear in increasing order, with the odd numbers appearing once and the even numbers appearing twice 1,0,2,0,4,0,8,0,16,0,… The even-numbered terms are all 0; the odd-numbered terms are successive powers of 2

8 Arithmetic Progression An arithmetic progression is a sequence of the form a, a+d, a+2d, a+3d, …, a+nd For example, the sequence: 1, 7, 13, 19, 25, 31, 37, 43, 49, 55 … is an arithmetic progression with a = 1 and d = 6 The next term in this arithmetic progression will be a + d(n-1)

9 Well-known Sequences n 2 = 1, 4, 9, 16, 25, … n 3 = 1, 8, 27, 64, 125, … n 4 = 1, 16, 81, 256, 625, … 2 n = 2, 4, 8, 16, 32, … 3 n = 3, 9, 27, 81, 243, … n! = 1, 2, 6, 24, 120, …

10 Summations This symbol represents the following sum: a m + a m+1 + … + a n where: j: subscript of term (index of summation) m: 1st subscript value (lower limit) n: last subscript value (upper limit) Note that the choice of these letters (a, j, m and n) is arbitrary

11 Example Express the sum of the first 100 terms of sequence {a n } where a n = n for n = 1, 2, 3, … Recall the summation denotation: So j (the index of summation) goes from m=1 to n=100, and we want the sum of all (j 2 + 1) between m and n Thus the expression is:

12 Example Find the value of each of the following summations: (1+1) + (2+1) + (3+1) + (4+1) + (5+1) = = 20 (-2) 0 + (-2) 1 + (-2) 2 + (-2) 3 + (-2) 4 = = 11

13 Geometric Progression A geometric progression is a sequence of the form ar 0, ar 1, ar 2, ar 3, ar 4, …, ar k where: –a = initial term –r = common ratio –both a & r are real numbers The summation of the terms of a geometric progression is called a geometric series

14 S: sum of the first n+1 terms of a geometric series To find S, we could do the problem longhand, but we can derive a formula that provides a significant shortcut by following the steps below: 1. Multiply both sides by r: 2. Shift the index of summation. Suppose k = j + 1; then: 3. Shift back to 0:Sincewas the original S, we can conclude rS = S + (ar n+1 - a) so if r  1, S = (ar n+1 - a)/(r-1) and if r = 1, S = (n+1)a

15 Finding S: Example Find the value of: In this expression, a = 3, r = 2 and n = 8 Applying the formula: S = (ar n+1 - a)/(r-1) S = (3 * ) / (2 - 1) = 3 * = 1533 Can confirm this by doing problem longhand: 3* * * * * * * *2 7 +3*2 8 = = 1533

16 Double Summations A double summation has the form of one sigma after another, followed by the formula involving the two indexes of summation To solve a double summation: 1. Expand the inner summation 2. Compute the outer summation given the expansion

17 Double Summation Example 1. Expand the inner summation (i*1 + i*2 + i*3) 6i 2. Compute the outer summation given the expansion = (6*1 + 6*2 + 6*3 + 6*4 + 6*5 = 90

18 Double Summation Example (2i+0) + (2i+3) + (2i+6) + (2i+9) (8i + 18) = = 78

19 Using Summation Notation with Sets & Functions   f(s) s  S represents the sum of all values f(s) where s  S   s= = 6 s  {1,2,3}   s 2 + s= = 20 s  {1,2,3}

20 Formulae for commonly- occurring summations  n  ar k (ar n+1 – a) / (r – 1), r  1 k=0 n  k(n(n + 1)) / 2 k=1 

21 Formulae for commonly- occurring summations n  k 2 (n(n + 1)(2n + 1)) / 6 k=1  n  k 3 (n 2 (n + 1) 2 ) / 4 k=1

22 Using the formulae to solve summation problems Sometimes summation problems don’t start at a convenient index. For example, find:   k=  k-  k k=100k=1 k=1 Applying formula: n  k(n(n + 1)) / 2 k=1 (200(201))/2 - (99(100))/2 = = 15150

23 Section 3.2 Sequences and Summations - ends -