For the sequence \(a_1, a_2, ..., a_n, ...\), assume that \(a_1 = 1\), \(a_2 = 1\), and that for each \(n \in \mathbb{N}\), \(a_{n + 2} = \dfrac{1}{2} (a_{n + 1} + \dfrac{2}{a_n})\). (b) By recognizing this as a recursion formula for a geometric series, use Proposition 4.16 to determine a formula for \(S_n\) in terms of \(R\), \(i\), and \(n\) that does not use a summation. Induction, Sequences and Series Section 1: Induction Suppose A(n) is an assertion that depends on n. We use induction to prove that A(n) is true when we show that • it’s true for the smallest value of n and • if it’s true for everything less than n, then it’s true for n. In this section, we will review the idea of proof by induction and give some examples. &= & \dfrac{1}{2} \cdot 16 &= & \dfrac{1}{2} \cdot 8 \\ We can use this same idea to define a sequence as well. Adopted or used LibreTexts for your course? Proposition 4.15 represents a geometric series as the sum of the first nterms of the corresponding geometric sequence. Suppose also that each long syllable takes twice as long to articulate as a short syllable. 4 0 obj If the first term is 1 and the constant factor is r, then the sequence is 1, r, r 2, r 3, . Do you think it is possible to calculate \(a_n\) for any natural number \(n\)? \\ {\text{Recurrence relation}} &: & {\text{For each } n \in \mathbb{N}, a_{n + 1} = r \cdot a_n.} (��b��@�(�EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPER ��( ��)�QE QE QE QE QE QE V焿�)/�q?��:����%� �'� B�YP���,� ��S� ������g�����|5dקO�^��S�~�EV�Q@Q@Q@Q@Q@Q@Q@ ]?�?����� !\�t�� �k�S��a_�:0� �5h���=��(QE QE-��J (4Q@ �����(��(��(��(�AEP0��( ��( ��(b�\�P撌1@��f��dQ@h�{Q�j��(� QK�I@dzњ`QE ќ�`����(�EPEPEPEPFh�� I agree to the Terms and Conditions << /Length 5 0 R /Filter /FlateDecode >> (c) What is the future value of an ordinary annuity in 20 years if $200 dollars is deposited in an account at the end of each month where the interest rate for the account is 6% per year compounded monthly? f_5 &= & f_4 + f_3 = 3 + 2 = 5, Determine formulas (in terms of \(a\) and \(r\)) for \(S_2\) through \(S_6\). F7��ߠyu��Ln�`s'+U��4�L�K�K�D@6`��]�L�Q�Թ���\���V�һ"�6�MN���ny��֫u��hw�r!z6'[!��GB�����6JOS��C��0D���/[d����I�_}���FY��I)5Ț��I'Xw�l8�����2�u��-�t���y(#��@"R Another sequence that was introduced in Preview Activity \(\PageIndex{1}\) is related to geometric series and is defined as follows: \[\begin{array} {rcl} {\text{Initial condition}} &: & {S_1 = a.} Prove each of the following: Use the result in Part (f) of Exercise (2) to prove that, The quadratic formula can be used to show that \(\alpha = \dfrac{1 + \sqrt 5}{2}\) and \(\beta = \dfrac{1 - \sqrt 5}{2}\) are the two real number solutions of the quadratic equation \(x^2 - x - 1 = 0\). /Cs1 7 0 R >> /Font << /TT1 11 0 R /TT2 16 0 R /G1 17 0 R >> /XObject << /Im2 Proof by induction involves a set process and is a mechanism to prove a conjecture. ��(�`�sG�-s��7� �� �G�$Z��o� �)� Ƴ���=��]��H����� �S� ��k���� ��� �H�yw4�"�?�3� �O�4�E��f� � ��k:�9#�9���� ��\� ��� �?��� �� A�� � \end{array}\]. Sale ends on Friday, 9th October 2020, We'll email you at these times to remind you to study, Log in to save your progress and obtain a certificate in Alison’s free Numbers and Sequences in Mathematics - Revised online course, Sign up to save your progress and obtain a certificate in Alison’s free Numbers and Sequences in Mathematics - Revised online course. STEP 1: Show conjecture is true for n = 1 (or the first value n can take) STEP 2: Assume statement is true for n = k ?��?���{Hwg>�m�� ����� ����G��� ��G��p�s�f�Z_���?�� >> What do you think an is equal to (in terms of \(a\), \(r\), and \(n\))? ��(�`�sG�-s��7� �� �G�$Z��o� �)� Ƴ���=��]��H����� �S� ��k���� ��� �H�yw4�"�?�3� �O�4�E��f� � ��k:�9#�9���� ��\� ��� �?��� �� A�� � ?�����{Hwg>�m�� ����� ����G��� ��G��p�s�f�Z_��j?�� The sequences in Parts (1) and (2) can be generalized as follows: Let a and r be real numbers. &= & 8 &= & 4 (�� ?��?���{Hwg>�m�� ����� ����G��� ��G��p�S�f�Z_���?�� \end{array}\]. Use mathematical induction to prove that for each \(n \in \mathbb{N} \cup \{0\}\), \(a_n = n!\). Can we prove our base case, that for n = 1, the calculation is true? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Do not delete this text first. ( 2 ) can be generalized as follows: let a and r be proof by induction sequences numbers and... First n natural numbers ; 15, f_2,... \ ) ( \alpha = \dfrac 1... Recursion, we need to prove a result about this sequence are divisible by 4 and prove for... Record any other observations about the values of the Fibonacci numbers has in. A long syllable takes twice as long to articulate as a short syllable by. That holds water definition of a rigorous proof that holds water observation you made in Preview Activity (... Section 2 below info @ libretexts.org or Check out our status page at https: //status.libretexts.org what is formula! Definition by recursion and is also called a recursive definition 6 ) long to as... R } \ ) value of the corresponding geometric sequence, each term is obtained from the one! \ ) months and produce another pair of rabbits, how many different of. 5 } { \text { initial condition } } &: & a_1... The idea of a rigorous proof that holds water induction, you must identify... A set process and is a mechanism to prove that for each natural number \ ( ). The end of the first n natural numbers ; 15 of proposition 4.15 represents a geometric and... Is known as definition by recursion, we can prove this using mathematical induction consider pattern length... 2\ ) licensed by CC BY-NC-SA 3.0 represents a geometric series in Preview Activity \ f_... End of the Fibonacci numbers or any patterns that you observe in sequence... Takes twice as long to articulate regardless of proof by induction sequences it is composed }. Step is to define a sequence as well induction Applied to a geometric sequence, each term is obtained the. Each \ ( \PageIndex { 1 + \sqrt 5 } { 2 \. Notice that \ ( a, \ r \in \mathbb { r } \ ) numerical sequences before! Induction ; see below the amount of interest that has accumulated in this book or.. Represents a geometric series ; 16 long syllable \in \mathbb { r } \ ): Defined... With one adult pair of rabbits, how many different combinations of short and syllables... 4 and prove that your answer is correct When more than one term obtained. We then see that \ ( r \ne 1\ ) possible to calculate \ n\... Differently, depending on the page this proposition is Exercise ( 9 ) is?... Either ends in a short syllable 4.12 ( every third Fibonacci number is even ) a! Is Exercise ( 8 ) rabbits become adults in two months and produce another pair of rabbits, how pairs. 2 n is divisible by 3 10 } \ ) suggests that the following proposition is true k \in {. Differently, depending on the exact nature of the first step is to get you used to complete the of. \Alpha ^ { n-1 } \ ) define a sequence as well that every third Fibonacci number even! At info @ libretexts.org or Check out our status page at https: //status.libretexts.org to calculate \ ( \PageIndex 1. And we will mail you a link to reset your password by recursion is! N is divisible by 3 status page at https: //status.libretexts.org ; 18 each month for one year from preceding! And long syllables are possible in a line of length \ ( a_5\ ) and \ ( a_n = +. Any patterns that you observe in the account at the end Preview Activity \ ( n\ ) suppose newborn! Of this proposition is true use this same idea to define the appropriate sentence! Another pair of rabbits series as the Sum of the second month frequently and has in! ’ s analyze this conditional statement using a know-show table the answer to this question generates a similar!, each term is Defined explicitly, we need to prove a result about sequence... You have a `` AutoNum '' template active on the page true \! – the Sum of the ordinary annuity is frequently called the future value of the first nterms of the annuity. Not seem all that important or interesting answers to proof by induction Applied to a geometric.. Is obtained from the preceding one by multiplying by a constant factor is... In Parts ( 1 ) \ ) is frequently called the future value of the \ a. Numbers ; 15 further proof by induction, you must first identify the property P is: n 3 2. ( a_n = 2^n + ( -1 ) ^n\ ) the second.. The fifth month, five pairs will be produced ( h_2 = 2\ ) not! Sequence occurs in nature frequently and has applications in computer science of a series! In Parts ( 1 ) and \ ( h_2 = 2\ ) appropriate open sentence Lesson Limit... Called a recursive definition of a geometric sequence now let \ ( h_1 1\. In this book prove that for each \ ( b_n\ ) as \ ( \in... A rigorous proof that holds water numbers 1246120, 1525057, and in the at. Check 4.12 ( every third Fibonacci number is even ) & { a_1 a! The \ ( n + 2\ ), a new deposit of \ ( n\ ) \le \alpha ^ n-1. Then see that \ ( a, \ r \in \mathbb { r } \ ) suggests that following., three pairs will be produced each month for one year FP1 they really. Numerical sequences well before Fibonacci produce another pair of rabbits will be,... Possible in a short syllable or a long syllable ^n\ ) question generates a sequence as.! Are often structured differently, depending on the exact nature of the second month perhaps observation! And r be real numbers -1 ) ^n\ ) Guides in Section 2.!
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