1.1 Generalities on sequences

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UNIT 1 SEQUENCES AND SERIES Key unit competence: Apply arithmetic and geometric sequences to solve problems in financial mathematics. 1.0 INTRODUCTORY ACTIVITY Suppose that an insect population is growing in such a way that each new generation is 2 times as large as the previous generation. If there are 126 insects in the first generation, on a piece of paper, write down the number of insects that will be there in second, third, fourth,…nth generation. 1.1 Generalities on sequences ACTIVITY 1.1 Fold once an A4 paper, what is the fraction that represents the part you are seeing? Fold it twice, what is the fraction that represents the part you are seeing? What is the fraction that represents the part you are seeing if you fold it ten times? What is the fraction that represents the part you are seeing if you fold it n times? Write a list of the fractions obtained starting from the first until the nth fraction. 12 CONTENT SUMMARY Let us consider the following list of numbers: 11111,,,,...,... 234n . The terms of this list are compared to the images of the function1()fxx= . The list never ends, as the ellipsis indicates. The numbers in this ordered list are called the terms of the sequence. In dealing with sequences, we usually use subscripted letters, such as1u to represent the first term, 2u for the second term, 3u for the third term, and so on such as in the sequence1()nfnun== . However, in the sequence such as {}:3,nnuun=− the first term is 3u as the previous are not possible, in the sequence {}:25,nnuun=− the first term is 0u . Definition A sequence is a function whose domain is the set of natural numbers. The terms of a sequence are the range elements of the function. It is denoted by 1231,,,...,,nnuuuuu− and shortly {}nu . We can also write 1231{,,,...,,}nnuuuuu− . The dots are used to suggest that the sequence continues indefinitely, following the obvious pattern. The numbers 1231,,,...,,nnuuuuu− in a sequence are called terms of the sequence. The natural number n is called term number and value nu is called a general term of a sequence and the term 1u is the initial term or the first term. As a sequence continues indefinitely, it can be denoted as {}1nnu∞ = . The number of terms of a sequence (possibly infinite) is called the length of the sequence. Notice • Sometimes, the term number, n, starts from 0. In this case terms of a sequence are 0121,,,...,,,...nnuuuuu− and this sequence is denoted by {}0nnu+∞ = . In this case the initial term is 0u . • A sequence can be finite, like the sequence2,4,8,16,...,256 . The empty sequence {} is included in most notions of sequences, but may be excluded depending on the context. Usually a numerical sequence is given by some formula()nufn= , permitting to find any term of the sequence by its number n; this formula is called a general term formula. A second way of defining a sequence is to assign a value to the first (or the first few) term(s) and specify the nth term by a formula or equation that involves one or more of the terms preceding it. Sequences defined this way are said to be defined recursively, and the rule or formula is called a recursive formula. Example 1: The sequence 111,.nnuunu−== Finite and infinite sequences: Consider the sequence of odd numbers less than 11: This is 1, 3,5,7,9. This is a finite sequence as the list is limited and countable. However, the sequence made by all odd numbers is: 1,3,5,7,9,...21,...n+ This suggests the definition that an infinite sequence is a sequence whose terms are infinite and its domain is the set of positive integers. Note that it is not always possible to give the numerical sequence by a general term formula; sometimes a sequence is given by description of its terms. Examples 2: 1) Numerical sequences: 0, 1,2,3,4,5,... a sequence of natural numbers; 0,2,4,6,8,10..... a sequence of even numbers; 1.4,1.41,1.414,1.4142,... a numerical sequence of approximate, defined more precisely by the values of2 . For the last sequence it is impossible to give a general term formula, nevertheless this sequence is described completely. 2) List the first five term of the sequence {}12nn+∞ = Solution Here, we substitute 1,2,3,4,5,...n= into the formula 2n . This gives 123452,2,2,2,2,... Or, equivalently, 2,4,8,16,32,... 3) Express the following sequences in general notation a) 1234,,,,... 2345 b) 1111,,,,... 24816 Solution a) 1234,,,,... 2345 Beginning by comparing terms and term numbers: Term number 1 2 3 4 … Term 12 23 34 45 … In each term, the numerator is the same as the term number, and the denominator is one greater the term number. Thus, the thn term is 1nn+ and the sequence may be written as 11nnn+∞ =  + . b) 1111,,,,... 24816 Beginning by comparing terms and term numbers: Term number 1 2 3 4  Term 12 14 18 116  Or Term number 1 2 3 4  Term 112 212 312 412  In each term, the denominator is equal to 2 powers the term number. We observe that the thn term is 12n and the sequence may be written as 112nn+∞ =    . 4) A sequence is defined by {}011: 32nnnuuuu+ = =+ Determine123,uuandu Solution Since 01u= and 132nnuu+=+ , replace n by 0,1,2 to obtain 123,,uuu respectively. 01100,323125nuuu+===+ =×+ = ⇒ 11211,3235217nuuu+===+ =×+ = 21322,32317253nuuu+===+ =×+ = ⇒ Thus, 12351753uuu= = = APPLICATION ACTIVITY 1.1 1. A sequence is given by Determine and 2. List the first five term of the sequence 3. Express the following sequence in general notation 1, 3, 5, 7, 9, 11, …