12.112.3_papa2_class_notes
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http://www.houstonchristian.org/data/files/news/ClassHomework/12.112.3_PAPA2_Class_Notes.pdfTRANSCRIPT
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12.1 - 12.3Sequence
andSeries
Crash Course
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Sequence : A function f whose domain is the set of natural numbers. The values f(1), f(2), f(3), … are called terms of the sequence. So, a sequence is an ordered list of numbers, with each number being a term of the sequence.
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Note: · The domain of a sequence is the set of natural numbers, and the range is the terms of the sequence.
· That is, 1 is paired with 2, since 2 is the 1st
term of the sequence, hence the point (1,2)
· 2 is paired with 5, since the 2nd
term of the sequence is 5, hence the point (2,5)
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Recursive Form of a Sequence:A recursive formula is one in which each succeeding term is formulated from one or more previous terms.
One of the most famous sequences, one that has many recurrences in nature, music and elsewhere, is the Fibonacci sequence. See if you can determine its pattern and determine its recursive formula:
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A partial sum occurs when terms of a sequence are added together and it has its own notation:
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To find the next term in an arithmetic sequence, find the common difference by subtracting two consecutive terms, and add the difference to the last term that you know.
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A recursive formula is one in which each succeeding term is formulated from one or more previous terms. The n
th term of an arithmetic sequence can be found by
using such a formula where the 1st term and common difference are known.
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(each term gettingbigger, so a.must be prettysmall)
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10. Karmen's mother started a college savingsaccount for her sixth birthday. Shedeposited $25 the first month and thenincrease the deposit by $5 each month.How much will she have deposited intwelve years?
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A geometric sequence differs from an arithmetic sequence in that successive (or consecutive) terms are found by using a common ratio, denoted as r, rather than a common difference. It has the form:
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A geometric sequence can also be found by a recursive formula where the 1st term
and common ratio are known:
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EXAMPLE 13: The second term of a geometric sequence is 6 and the 6th term is
96. Find the 4th term of the sequence.
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INFINITE SERIES : (reminder, a series is the SUM of the terms of a sequence) An infinite series is the sum of the terms of an infinite sequence. Right now, in this course, we will only consider infinite geometric sequences, but in Calculus 2 (or BC, if you take it in high school), you will study many types of series.
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Do you recall that repea.ng decimal is a ra.onal number? This is because it MUST be able to be expressed as a FRACTION!!! A repea.ng decimal is an Infinite Geometric Series!!!
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