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7.2 Analyze Arithmetic 7.2 Analyze Arithmetic Sequences & Series Sequences & Series p.442 p.442 What is an arithmetic sequence? What is an arithmetic sequence? What is the rule for an arithmetic What is the rule for an arithmetic sequence? sequence? How do you find the rule when given How do you find the rule when given two terms? two terms?

7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for an arithmetic sequence? How do you find the rule when

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Text of 7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for...

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  • 7.2 Analyze Arithmetic Sequences & Series p.442 What is an arithmetic sequence? What is the rule for an arithmetic sequence? How do you find the rule when given two terms?
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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). Find the common difference by subtracting the term on the left from the next term on the right.
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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
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  • Example: Write a rule for the nth term of the sequence 32,47,62,77,. Then, find a 12. There is a common difference where d=15, therefore the sequence is arithmetic.There 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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  • One term of an arithmetic sequence is a 19 = 48. The common difference is d = 3. a n = a 1 + (n 1) d a 19 = a 1 + (19 1) d 48 = a 1 + 18(3) Write general rule. Substitute 19 for n Solve for a 1. a. So, a rule for the n th term is: a. Write a rule for the nth term. 6 = a 1 Substitute 48 for a 19 and 3 for d. SOLUTION a. Use the general rule to find the first term. a n = a 1 + (n 1) d = 6 + (n 1) 3 Write general rule. Substitute 6 for a 1 and 3 for d. Simplify.
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  • b. Graph the sequence. One term of an arithmetic sequence is a 19 = 48. The common difference is d =3. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. b.
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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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  • Two terms of an arithmetic sequence are a 8 = 21 and a 27 = 97. Find a rule for the nth term. SOLUTION STEP 1 Write a system of equations using a n = a 1 + (n 1)d and substituting 27 for n (Eq 1) and then 8 for n (Eq 2). STEP 2 Solve the system. 76 = 19d 4 = d 97 = a 1 + 26(4) Subtract. Solve for d. Substitute for d in Eq 1. 27 = a 1 Solve for a 1. STEP 3 Find a rule for a n. a n = a 1 + (n 1)d Write general rule. = 7 + (n 1)4 Substitute for a 1 and d. = 11 + 4n Simplify. a 27 = a 1 + (27 1)d 97 = a 1 + 26d a 8 = a 1 + (8 1)d 21 = a 1 + 7d Equation 1 Equation 2
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  • What is an arithmetic sequence?What is an arithmetic sequence? The difference between consecutive terms is a constant What is the rule for an arithmetic sequence?What is the rule for an arithmetic sequence? a n =a 1 +(n-1)d How do you find the rule when given two terms?How do you find the rule when given two terms? Write two equations with two unknowns and use linear combination to solve for the variables.
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  • 7.2 Assignment p. 446, 3-35 odd
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  • Analyze Arithmetic Sequences and Series day 2 What is the formula for find the sum of a finite arithmetic series?
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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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  • SOLUTION a 1 = 3 + 5(1) = 8 a 20 = 3 + 5(20) =103 S 20 = 20 ( ) 8 + 103 2 = 1110 Identify first term. Identify last term. Write rule for S 20, substituting 8 for a 1 and 103 for a 20. Simplify. ANSWER The correct answer is C.
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  • You are making a house of cards similar to the one shown Write a rule for the number of cards in the nth row if the top row is row 1. a. House Of Cards SOLUTION Starting with the top row, the numbers of cards in the rows are 3, 6, 9, 12,.... These numbers form an arithmetic sequence with a first term of 3 and a common difference of 3. So, a rule for the sequence is: a. a n = a 1 + (n 1) = d = 3 + (n 1)3 = 3n Write general rule. Substitute 3 for a 1 and 3 for d. Simplify.
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  • You are making a house of cards similar to the one shown What is the total number of cards if the house of cards has 14 rows? b. House Of Cards SOLUTION Total number of cards = S 14 Find the sum of an arithmetic series with first term a 1 = 3 and last term a 14 = 3(14) = 42. b.
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  • 5. Find the sum of the arithmetic series (2 + 7i). i = 1 SOLUTION a 1 = 2 + 7(1) = 9 a 12 = 2 + (7)(12) = 2 + 84 = 86 ( ) S n = n a 1 + a n 2 S 12 = 570 ANSWER S 12 = 570
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  • What is the formula for find the sum of a finite arithmetic series?
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  • 7.2 Assignment: p. 446 40-48 all, 63-64