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Obj.: I will be able to identify an infinite series. I will be able to read and write recursion formulas. I will be able to graph sequences.VocabularyFibonacci sequenceGeneral TermRecursion FormulasFinite SequenceGraph of a sequenceInfinite SequenceNotes Sequences-65722510586600Many creations in nature involve ______________ mathematical designs, including a variety of ______________.The _________________ sequence can be found in nature in many places. The sequence begins with two _______ and every term thereafter is the ______ of the _______ ________________ _________________________________________________________________ is a ______________ term of the sequence, also called the _______ __________.An _____________ sequence is a ____________ whose domain is the set of _____________ ____________. The function values, or terms, of the sequence are represented by________________________________Sequences whose domains consist only of the _________ _____ positive integers are called ________________ _______________________.Because a sequence is a function whose domain is the set of positive integers, the graph of a sequence is a set of ______________ ______________ (______ ___________________). __________________ FormulasA _____________________formula _____________ the nth term of a sequence as a _________________ of the ____________________ ____________. Example: _______________________Recursion formulas can be used to find the ___________ ____________ in a sequence, but you must already know the previous term in order to find it.PracticeWrite the first four terms of the sequence whose general term is given.1.an=5n+3 2.an=8n-5 3.an=-1n(n+1) 4.an=-2n(n+3) The sequence given is defined using a recursion formula. Write the first four terms of the sequence.5.a1=7 and an=an-1+1 for n≥2 6.a1=9 and an=an-1-5 for n≥2 7.a1=4 and an=3an-1 for n≥2 8.a1=2 and an=4an-1+4 for n≥2 9. As n increases, the terms of the sequence an=1+1nn get closer and closer to the number e (where e≈2.7183). Use a calculator to find a10, a100, a1000, a10000, and a100000, comparing these terms to your calculator’s decimal approximation for e.Obj.: I will be able to read, write, and evaluate factorial notation. I will be able to read, write, and manipulate summation notation for the sums of a sequence.VocabularyFactorial NotationSummation NotationIndex of SummationUpper Limit of SummationLower Limit of SummationExpanding the Summation NotationNotes_________________ NotationProducts of _____________________ positive integers occur quite often in sequences. Factorial notation is a special notation for ____________________ ________________.If ___ is a positive integer, the notation _____ (Read “___ ________________”) is the ________________of all positive integers from ___ down through ___. ___________________________________________________ by definitionLike exponents, factorials only affect the number they are ____________ ______________ unless grouping symbols, like ___________________, appear. _____________________________________________________________________________________________ NotationIt can be useful to find the sum of the first n terms of a sequence. If there are too many terms, writing out the entire sum can be cumbersome. Summation notation can be used to simplify writing sums of a sequence. The sum of the first n terms of a sequence is represented by _____ is the __________ of _________________, which indicates the ______________ of the terms of the sequence being _______________.______ is the _____________ _________ of summation (the _____________ value of the index)_______ is the ______________ __________ of the summation (the ________________ value of the index)NOTES:________ ____________ can be used for the indexThe __________ limit can be ___________ ______________ than 1Writing out the sum of the terms is called __________________ the summation notationProperties of Sums PracticeWrite the first four terms of the sequence whose general term is given below:1.an=n4n-1! 2.an=5n+3! Evaluate the factorial expression.3.12!2!10! 4.14!3!11! Find the indicated sum.5. j=169j6.j=165j7.i=15i!i-1!8.i=15i(i+1)Express the sum using summation notation. Use the lower limit of the summation given and k for the index of summation.9.5+8+11+14+…+47 k=110.4+7+10+!3+…+46 k=111.2a+2a+d+2a+2d+2a+3d+…+(2a+nd) k=0Obj.: I will be able to recognize an arithmetic sequence and graph its function. I will be able to find the common difference fro an arithmetic sequence and give a set of values for the sequence.VocabularyArithmetic SequenceCommon DifferenceGeneral Term of Arithmetic SequenceNotes Arithmetic SequencesAn______________________ sequence is a sequence in which each term after the first ____________ from the preceding term by a _________________ _______________.The difference between ___________________ terms is called the _________________ __________________ of the sequence.Finding the Common Difference for an Arithmetic SequenceRepresented by the symbol ____.________________ any _________ ____________________ _____________ in an arithmetic sequence to find the common difference.An arithmetic sequence is a ____________ function whose ______________ is the set of _________________ _______________The graph of this function is a set of ______________ _____________ (points not connected)The points lie on a ________________ ______________.Writing the Terms of an Arithmetic Sequence To write a set of terms in an arithmetic sequence, you must know at least ________ _________, usually a1, and the ___________________ _____________________.Use the _________________ formula _____________________PracticeWrite the first six terms of the arithmetic sequence with the given first term, a1, and common difference, d.1.a1=300 d=30 2.a1=-9 d=4 3.a1=12 d=12 4.a1=210 d=-60 Write the first six terms of the arithmetic sequence shown below.5.an=an-1+8, a1=-7 6.an=an-1-18, a1=34 7.an=an-1-0.3, a1=0.6 8.an=an-1-0.7, a1=2.8 Look at the graph of the arithmetic sequence {bn}. If {bn} is a finite sequence whose last term is given below, how many terms does {bn} contain?9.Last term is 11410.Last term is -125Obj.: I will be able to find a specific term within an arithmetic sequence. I will be able to find the sum of the first n terms of an arithmetic sequence.VocabularySum of the First n Terms of Arithmetic SequenceNotes The General term of an Arithmetic Sequence Consider how the ____________ ________ ____________ in an arithmetic sequence are determined: The nth term (____________ _____________) of an arithmetic sequence with the first term _____ and common difference ____ is The Sum of the First n Terms of an Arithmetic SequenceThe sum of the first n terms of an arithmetic sequence, denoted by Sn and called the _____ ______________ _________, can be found _____________ having to _______ ______ all the terms.The sum, _____, of the first ____ terms of an arithmetic sequence is given by PracticeUse the formula for the general term of an arithmetic sequence to find the indicated term of the sequence with the given first term, a1, and common difference, d.1.a100 when a1=-50, d=5 2.a100 when a1=-40, d=5 Write a formula for the general term of the arithmetic sequence. Then use the formula for an to find a20.3.4, 1, -2 ,-5,… 4.5,3,1,-1,… Find the sum of the first n terms of the sequence below.5.n=25 for sequence 5, 9, 13, 17,…6.n=50 for sequence-16,-8,0,8,… 7.n=140 sequence: positive even integers8.n=180 sequence: positive multiples of 3Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.9.i=150(-8i+9)10.i=110(-4i+3)11.Use a system of two equations in two variables, a1 and d, to write a formula for the general term of the arithmetic sequence whose seventh term, a7, is 2 and whose ninth term, a9, is 8.pany A pays $25000 yearly with raises of $1600 per year. Company B pays $28000 yearly with raises of &700 per year. Which company will pay more in year 10? How much more?Obj.: I will be able to recognize geometric sequences. I will be able to find the common ratio and a set of terms for a geometric sequence. I will be able to find the general term of a geometric sequence.VocabularyGeometric SequenceCommon RatioGeneral Term of a Geometric SequenceNotesGeometric SequencesA _____________ sequence is a sequence in which each term after the first is obtained by ___________________ the preceding term by a ________ _____________ _________________. The amount by which we multiply each term is called the _________________ ___________ of the sequence.Find the Common Ratio of a Geometric SequenceThe _______________ __________ ___ is found by ____________ any term after the first term by the term that directly ________________ it. A geometric sequence with a ________________ _______________ _________ other than ____ is an _______________ function whose _____________ is the ______________ set of __________________.Write Terms of a Geometric SequenceTo ____________ ___ _________ of terms in a geometric sequence, you must know at least one term, usually ______, as well as the __________________ ___________.Use the recursion formula: ____________________The General Term of a Geometric SequenceThe nth term (the _____________ _________) of a geometric sequence with the first term _____ and the common ratio ____ isPracticeWrite the first five terms of the geometric sequence with the first term and common ratio provided.1.a1=7, r=3 2.a1=32, r=12 Write the first five terms of the geometric sequence.3.an=-6an-1, a1=-2 4.an=4an-1, a1=1 Use the formula for the general term of a geometric sequence to find the indicated term of the following sequence with the given first term and common ratio.5.Find a10 when a1=6000000, r=0.1 6.Find a8 when a1=5, r=-2Write a formula for the nth term of the following geometric sequence. Then use the formula for an to find a7.7.2, 6, 18,… 8.0.0003, -0.003, 0.03, -0.3,… Find a2 and a3 for the following geometric sequence.9.27, a2 , a3, 64 10.250, a2 , a3, 16 Obj.: I will be able to find the sum of n terms of a geometric sequence. I will be able to understand and find values of an annuity. I will be able to use geometric series and their sums to find multiplier effect values on the economy.Vocabularynth Partial SumAnnuityValue of the AnnuityMultiplier EffectInfinite Geometric SeriesSum of Infinite Geometric SeriesNotesThe Sum of the First n Terms of a Geometric SequenceThe _______ of the first ___ __________ of a geometric sequence, denoted by _____ and called the ______ _____________ _______, can be found without having to add up all the terms. The sum of the first n terms of a geometric sequence is given by The common ratio ______________ _____________ ____.AnnuitiesAn ___________ is a sequence of __________ payments made at equal time periods.An ________ is an example of an annuity.The terms in this sequence are based on the simple interest formula: _______________The value of the annuity is the ______ of all _____________ made plus all ______________ paid.If P is the deposit made at the end of each compounding period for an annuity at r percent annual interest compounded n times per year, the value, A, of the annuity after t years is Geometric SeriesAn ______________ sum in the form with the first term a1 and common ratio r is called an infinite ______________ _________.The sum of an Infinite Geometric SeriesIf -1<r<1, then the sum of the infinite geometric series above is given by If |r|≥1, the infinite series does not have a sum.________________ __________A _____ __________ that returns a certain amount of money to taxpayers can have a total effect on the economy that is _________ __________ this amount. In economics, this _____________________ is called the multiplier effect. This can be found by finding the sum of an infinite geometric series. PracticeFind the sum of the first 11 terms of the geometric sequence. Use the formula for the sum of the first n terms of a geometric sequence.1.3,-6,12,-24, … 2.6, -30, 150, -750, … Find the indicated sum.3.i=195*2i4.i=1114*2iFind the sum of the infinite geometric series.5.5-56+536-5216+… 6.4-47+449-4343+… 7.i=1∞80.8i-18.i=1∞23-0.1i-1If {an} and {bn} equal the following, find a10+b109.an=-3, 6, -12, 24, …, bn=6,-11,-28,-45, …, 10.an=-6, 12,-24,48, …, bn=15, 4, -7, -28, …, 11.At age 30, to save for retirement, Rebecca decides to deposit $100 at the end of each both in an IRA that pays 5.7% compounded monthly. Use the formula for the value of an annuity. a. How much will she have from the IRA when she retires at age 65? b. Find the interest.12.A new factory in a small town has an annual payroll of $2 million. It is expected that 50% of this money will be spent in the town by factory personnel. The people in this town who receive this money are expected to spend 50% of what they receive in town, and so on. What is the total of all this spending, called the total economic impact of the factory, on the town each year? Use the formula or the sum of an infinite geometric series. ................
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