7.2 Analyze Arithmetic Sequences

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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.

• 5,11,17,23,29,
• 11-56
• 17-116
• 23-176
• 29-236
• Arithmetic (common difference is 6)

• -10,-6,-2,0,2,6,10,
• -6--104
• -2--64
• 0--22
• 2-02
• 6-24
• 10-64
• Not arithmetic (because the differences are not
  the same)

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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 a12.

• There is a common difference where d15,
  therefore the sequence is arithmetic.
• Use ana1(n-1)d
• an32(n-1)(15)
• an3215n-15
• an1715n
• a121715(12)197

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One term of an arithmetic sequence is a19
48. The common difference is d 3.
a.
a. Write a rule for the nth term.
SOLUTION
a. Use the general rule to find the first term.
an a1 (n 1) d
Write general rule.
a19 a1 (19 1) d
Substitute 19 for n
Substitute 48 for a19 and 3 for d.
48 a1 18(3)
Solve for a1.
6 a1
So, a rule for the nth term is
Write general rule.
an a1 (n 1) d
6 (n 1) 3
Substitute 6 for a1 and 3 for d.
 
Simplify.
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b. Graph the sequence. One term of an arithmetic
sequence is a19 48. The common difference is d
3.
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Example One term of an arithmetic sequence is
a850. The common difference is 0.25. Write a
rule for the nth term.

• Use ana1(n-1)d to find the 1st term!
• a8a1(8-1)(.25)
• 50a1(7)(.25)
• 50a11.75
• 48.25a1
• Now, use ana1(n-1)d to find the rule.
• an48.25(n-1)(.25)
• an48.25.25n-.25
• an48.25n

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Now graph an48.25n.

• Just like yesterday, remember to graph the
  ordered pairs of the form (n,an)
• 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
a510 and a30110. Write a rule for the nth term.

• Begin by writing 2 equations one for each term
  given.
• a5a1(5-1)d OR 10a14d
• And
• a30a1(30-1)d OR 110a129d
• Now use the 2 equations to solve for a1 d.
• 10a14d
• 110a129d (subtract the equations to cancel a1)
• -100 -25d
• So, d4 and a1-6 (now find the rule)
• ana1(n-1)d
• an-6(n-1)(4) OR an-104n

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Example (part 2) using the rule an-104n, write
the value of n for which an-2.

• -2-104n
• 84n
• 2n

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Two terms of an arithmetic sequence are a8 21
and a27 97. Find a rule for the nth term.
SOLUTION
Equation 1
Equation 2
Subtract.
76 19d
Solve for d.
4 d
Substitute for d in Eq 1.
97 a1 26(4)
27 a1
Solve for a1.
an a1 (n 1)d
Write general rule.
Substitute for a1 and d.
7 (n 1)4
11 4n
Simplify.
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• What is an arithmetic sequence?
• The difference between consecutive terms is a
  constant
• What is the rule for an arithmetic sequence?
• ana1(n-1)d
• 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 sequence
• The formula to find the sum of a finite
  arithmetic series is

Last Term
1st Term
of terms
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Example Consider the arithmetic series
20181614 .

• Find the sum of the 1st 25 terms.
• First find the rule for the nth term.
• an22-2n
• So, a25 -28 (last term)

• Find n such that Sn-760

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• -1520n(2022-2n)
• -1520-2n242n
• 2n2-42n-15200
• n2-21n-7600
• (n-40)(n19)0
• n40 or n-19
• Always choose the positive solution!

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SOLUTION
a1 3 5(1) 8
Identify first term.
Identify last term.
a20 3 5(20) 103
Write rule for S20, substituting 8 for a1 and 103
for a20.
1110
Simplify.
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You are making a house of cards similar to the
one shown
SOLUTION
an a1 (n 1) d
Write general rule.
Substitute 3 for a1 and 3 for d.
3 (n 1)3
Simplify.
3n
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You are making a house of cards similar to the
one shown
SOLUTION
Find the sum of an arithmetic series with first
term a1 3 and last term a14 3(14) 42.
b.
 
Total number of cards S14
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5. Find the sum of the arithmetic series
(2 7i).
i 1
SOLUTION
a1 2 7(1) 9
a12 2 (7)(12) 2 84
86
 
 
S12 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