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11.2 Arithmetic Sequences & Series p.659

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Arithmetic Sequence: The difference between consecutive terms is constant (or the same). The constant difference is also known as the common difference (d). (It’s also that number that you are adding everytime!)

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Example: Decide whether each sequence is arithmetic. -10,-6,-2,0,2,6,10,…-10,-6,-2,0,2,6,10,… -6--10=4-6--10=4 -2--6=4-2--6=4 0--2=20--2=2 2-0=22-0=2 6-2=46-2=4 10-6=410-6=4 Not arithmetic (because the differences are not the same) 5,11,17,23,29,…5,11,17,23,29,… 11-5=611-5=6 17-11=617-11=6 23-17=623-17=6 29-23=629-23=6 Arithmetic (common difference is 6)Arithmetic (common difference is 6)

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Rule for an Arithmetic Sequence a n =a 1 +(n-1)d

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Example: Write a rule for the nth term of the sequence 32,47,62,77,…. Then, find a 12. The is a common difference where d=15, therefore the sequence is arithmetic.The is a common difference where d=15, therefore the sequence is arithmetic. Use a n =a 1 +(n-1)dUse a n =a 1 +(n-1)d a n =32+(n-1)(15) a n =32+(n-1)(15) a n =32+15n-15 a n =32+15n-15 a n =17+15n a n =17+15n a 12 =17+15(12)=197

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Example: One term of an arithmetic sequence is a 8 =50. The common difference is 0.25. Write a rule for the nth term. Use a n =a 1 +(n-1)d to find the 1 st term!Use a n =a 1 +(n-1)d to find the 1 st term! a 8 =a 1 +(8-1)(.25) 50=a 1 +(7)(.25) 50=a 1 +1.75 48.25=a 1 * Now, use a n =a 1 +(n-1)d to find the rule. a n =48.25+(n-1)(.25) a n =48.25+.25n-.25 a n =48+.25n

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Now graph a n =48+.25n. Just like yesterday, remember to graph the ordered pairs of the form (n,a n )Just like yesterday, remember to graph the ordered pairs of the form (n,a n ) So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.So, graph the points (1,48.25), (2,48.5), (3,48.75), (4,49), etc.

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Example: Two terms of an arithmetic sequence are a 5 =10 and a 30 =110. Write a rule for the nth term. Begin by writing 2 equations; one for each term given.Begin by writing 2 equations; one for each term given. a 5 =a 1 +(5-1)d OR 10=a 1 +4d And a 30 =a 1 +(30-1)d OR 110=a 1 +29d Now use the 2 equations to solve for a 1 & d.Now use the 2 equations to solve for a 1 & d. 10=a 1 +4d 10=a 1 +4d 110=a 1 +29d (subtract the equations to cancel a 1 ) -100= -25d So, d=4 and a 1 =-6 (now find the rule) a n =a 1 +(n-1)d a n =-6+(n-1)(4) OR a n =-10+4n

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Example (part 2): using the rule a n =-10+4n, write the value of n for which a n =-2. -2=-10+4n8=4n2=n

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Arithmetic Series The sum of the terms in an arithmetic sequenceThe sum of the terms in an arithmetic sequence The formula to find the sum of a finite arithmetic series is:The formula to find the sum of a finite arithmetic series is: # of terms 1 st Term Last Term

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Example: Consider the arithmetic series 20+18+16+14+…. Find the sum of the 1 st 25 terms.Find the sum of the 1 st 25 terms. First find the rule for the nth term.First find the rule for the nth term. a n =22-2na n =22-2n So, a 25 = -28 (last term)So, a 25 = -28 (last term) Find n such that S n =-760Find n such that S n =-760

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-1520=n(20+22-2n) -1520=-2n 2 +42n 2n 2 -42n-1520=0 n 2 -21n-760=0 (n-40)(n+19)=0 n=40 or n=-19 Always choose the positive solution!

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Assignment

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