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Chapter 1: Sequences and Sets 1.1 Sequences

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Sequences What number comes next? 1, 2, 3, 4, 5, ____ 2, 6, 10, 14, 18, ____ 1, 2, 4, 8, 16, ____ 6 22 32

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What comes next? 2, 5, 10, 17, 26, 37, ____ 1, 2, 6, 24, 120, ____ 2, 3, 5, 8, 12, ____ 1, 1, 2, 3, 5, 8, 13, ____ 50 720 17 21

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The key to any sequence is to discover its pattern The pattern could be that each term is somehow related to previous terms The pattern could be described by its relationship to its position in the sequence (1 st, 2 nd, 3 rd etc…) You might recognize the pattern as some well known sequence of integers (like the evens, or multiples of 10). You might be able to do all three of these ways!

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2, 4, 6, 8, 10 … Can we define a given element in relation to previous elements. – The first element has to be 2 since it has no previous elements – The second element is 2 more than the first element – The third element is 2 more than the second element. – In fact, each subsequent element is just two more than the previous one.

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2, 4, 6, 8, 10 … Can we represent this sequence in relation to its position? – At position 1 the value is 2 – At position 2 the value is 4 – At position 3 the value is 6 – At position n the value is 2 * n

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2, 4, 6, 8, 10 … Do we recognize this sequence as something familiar? –Yes. It is the positive even numbers.

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Mathematical Notation When we want to refer to terms in a sequence we usually use lower case letters (a, b, …) followed by a subscript indicating the position in the sequence we are referring to. a 1 is the first term in a sequence a 2 is the second term in a sequence a n is the nth term in a sequence

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2, 4, 6, 8, 10 … What is a 1 ? What is a 3 ? What is a 5 ? What is a n if n = 4? What is a n-1 if n = 4? 2 6 10 8 6

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Recursive Formula A recursive formula for a sequence is one where each term is described in relation to a previous term (or terms)

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2, 4, 6, 8, 10 … Write each term in relation to its prior term (as a recursive formula) What is a 1 ? a 1 =2 (no prior terms) What is a 3 ? a 3 = a 2 + 2 What is a 5 ? a 5 = a 4 + 2 What is a n ? a n = a n-1 + 2

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Closed Formula A closed formula for a sequence is a formula where each term is described only by its relation to its position.

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2, 4, 6, 8, 10 … Write each term in relation to its position (as a closed formula) What is a 1 ? a 1 =1* 2 (no prior terms) What is a 3 ? a 3 = 3 * 2 What is a 5 ? a 5 = 5 * 2 What is a n ? a n = n * 2

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Let’s try another Consider the sequence 5, 9, 13, 17… Write a recursive formula for this sequence. a 1 =5 a n = a n-1 + 4 Write a closed formula for this sequence a n = 4n + 1

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And another 1, 3, 7, 15, 31, 63… 123456n 137153163? Making a table can help to get a feel for what’s going on.

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The recursive formula actually isn’t too bad. a 1 =1 a n = 2a n-1 + 1 The closed formula is a little bit harder this time.

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Now try your hand at these. Find both recursive and closed formulas for the following sequences. 2, 6, 10, 14, 18, ____ 1, 2, 4, 8, 16, ____ 1, 2, 6, 24, 120, ____

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2, 6, 10, 14, 18, ____ Recursive Formula Closed Formula

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Recursive Formula Closed Formula 1, 2, 4, 8, 16, ____

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Recursive Formula Closed Formula 1, 2, 6, 24, 120, ____

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Find a recursive formula from a closed formula First write out the first several terms so you can understand the sequence 12345 18152229

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Find a recursive formula from a closed formula Establish the starting point in the pattern and then determine how each value relates the prior value

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Find a closed formula from a recursive formula 12345 272267202 These tend to be harder but you still want to establish and understand the sequence

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Find a closed formula from a recursive formula At this point unless you are really clever we are kind of stuck, the pattern is not one you may recognize. Specialized methods are needed to complete these.

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Common Mathematical Notion Summation: A summation is just the sum of the terms in a sequence. If the terms in the sequence are – 1, 2, 3, 4, 5, 6 then the summation is – 1+2+3+4+5+6 = 21 If the terms in the sequence are – 1, 4, 9, 16, 25 then the summation is – 1+4+9+16+25 = 55

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Summation is a very common Idea Because it is so common, mathematicians have developed a shorthand to represent summations (also called sigma notation) This is what the shorthand looks like, on the next few slides we will dissect it a bit.

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Dissecting Sigma Notation The giant Sigma just means that this represents a summation

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Dissecting Sigma Notation The i=1 at the bottom just states where is the sequence we want to start. If the value was 5 then we would start the sequence at the 5 th position

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Dissecting Sigma Notation The n at the top just says to what element in the sequence we want to get to. In this case we want to go up through the nth item.

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Dissecting Sigma Notation The portion to the right of the sigma is the closed formula for the sequence you want to sum over.

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Dissecting Sigma Notation closed formula 1n So this states that we want to compute the closed formula for each element from 1 to n.

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Dissecting Sigma Notation closed formula The portion to the right of the sigma is the closed formula for the sequence you want to sum over.

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Dissecting Sigma Notation Thus our summation is 1 + 2 + 3 + … + n If I told you that n had the value of 5, then the summation would be 1 + 2 + 3 + 4 + 5 = 15

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Let’s try a few. Compute the following summations

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How would you write the following sums using sigma notation? 5+10+15+20+25+30+35+40 1+8+27+64+125+216

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So why are sequences important Identifying patterns is an essential tool for anyone Developing a vocabulary to represent and analyze these sequences is the key to speaking the language of mathematics.

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