11.2 Arithmetic Sequences

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11.2 Arithmetic Sequences Series

• By L. Kealii Alicea

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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).
• (Its also that number that you are adding
  everytime!)

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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

• ana1(n-1)d

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Example Write a rule for the nth term of the
sequence 32,47,62,77, . Then, find a12.

• The 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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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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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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Assignment
11.2 A (all) 11.2 B (1-25 odd, 26-27)