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Arithmetic Sequence Chapter 2, lesson C

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IB standard Students should know Arithmetic sequence and series; sum of finite arithmetic series; geometric sequences and series; sum of finite and infinite geometric series Examples of applications, compound interest and population growth Sigma notation

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Arithmetic Sequences An arithmetic sequence is a sequence in which each term differs from the pervious one by the same fixed number Example 2,5,8,11,14 5-2=8-5=11-8=14-11 etc 31,27,23,19 27-31=23-27=19-23 etc

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Algebraic Definition {U n } is arithmetic U n+1 – U n = d for all positive integers n where d is a constant (the common difference) “If and only if” {U n } is arithmetic then U n+1 – U n is a constant and if U n+1 – U n is constant the {U n } is arithmetic

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The General Formula U 1 is the 1 st term of an arithmetic sequence and the common difference is d Then U 2 = U 1 + d therefore U 3 = U 1 + 2d therefore U 4 = U 1 + 3d etc. Then U n = U 1 + (n-1)d the coefficient of d is one less than the subscript

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Arithmetic Sequence For arithmetic sequence with first term u 1 and common difference d the general term (or the nth term) is u n = u 1 + (n-1)d

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Examples #1 Consider the sequence 2,9,16,23,30… Show that the sequence is arithmetic Find the formula for the general term U n Find the 100 th term of the sequence Is 828, 2341 a member of the sequence?

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The middle term If a, b, c are any consecutive terms of an arithmetic sequence the b - a= c - b (equating common differences) therefore 2b= a+c therefore b = (a+c) / 2 Thus the middle term is the arithmetic mean (average) of terms on each side of it - Hence the name arithmetic sequence

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Example #2 Find k given that 3k+1 and -3 are consecutive terms of an arithmetic sequence

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Example #3 Find the general term U n for an arithmetic sequence given that U 3 = 8 and U 8 = -17 U n = U 1 + (n-1)d

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Example #4 Insert four numbers between 3 and 12 so that all six numbers are in arithmetic sequence.

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Homework Page 42-44 #1-9

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