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Sequences and Series
Chapter 10
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Chapter 10 Agenda:Arithmetic Sequences
Arithmetic Series
Geometric Sequences
Geometric Series
Recursive Formulas
Fibonacci and Other Special Sequences
Infinite Geometric Series
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Arithmetic Sequences
Return to Table of Contents
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Arithmetic Sequence a list of numbers that have a common difference between terms
Find the next three terms:
1, 4, 7, 10, . . .
5, 11, 17, 23, . . .
9, 5, 1, 3, . . .
To find the common difference, "d", subtract the first term from the second term. The difference between successive terms should be the same for the entire sequence.
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Arithmetic Sequences
Find d:
1, 4, 7, 10, . . .
5, 11, 17, 23, . . .
9, 5, 1, 3, . . .
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1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . .
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2 Find the next term in the arithmetic sequence: 8, 4, 0, 4, . . .
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3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . .
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4 Find the value of d in the arithmetic sequence: 10, 2, 14, 26, . . .
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5 Find the value of d in the arithmetic sequence: 8, 3, 14, 25, . . .
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Arithmetic SequencesAs we study sequences we need a way of naming the terms: a1 represents the first term,a2 represents the second term,a3 represents the third term,and so on in this manner.
If we were talking about the 8th term we would use a8 .
When we want to represent any random term, call it the nth term and use an.
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Arithmetic SequenceWrite the next three terms of the arithmetic sequence that is described.
a1 = 4; d = 6
a1 = 3; d = 3
a1 = 0.5; d = 2.3
a2 = 7; d = 5
a2 a3 a4
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6 Which sequence matches the description?
A 4, 6, 8, 10
B 2, 6,10, 14
C 2, 8, 32, 128
D 4, 8, 16, 32
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7 Which sequence matches the description?
A 3, 7, 10, 14
B 4, 7, 11, 13
C 3, 7, 11, 15
D 3, 1, 5, 9
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8 Which sequence matches the description?
A 7, 10, 13, 16
B 4, 7, 10, 13
C 1, 4, 7,10
D 3, 5, 7, 9
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Arithmetic Sequences
To find a specific term,say the 5th or a5 , you could write out all of the terms fairly quickly.
But what about the 100th term(or a100 )?
We need to find a formula to arrive at the answer quickly without writing out the whole list.
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Arithmetic Sequence
Consider: 3, 9, 15, 21, 27, 33, 39,. . .
a1 3
a2 9 = 3+6
a3 15 = 3+12 = 3+2(6)
a4 21 = 3+18 = 3+3(6)
a5 27 = 3+24 = 3+ 4(6)
a6 33 = 3+30 = 3+5(6)
a7 39 = 3+36 = 3+6(6)
Do you see a pattern that relates the term number to its
value?
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Arithmetic SequencesExample Find the 21st term of the arithmetic sequence with a1 = 4 and d = 3.
Example Find the 12th term of the arithmetic sequence with a1 = 6 and d = 5.
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Arithmetic SequencesExample Find the 1st term of the arithmetic sequence with a15 = 30 and d = 7.
Example Find the 1st term of the arithmetic sequence with a17 = 4 and d = 2.
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Arithmetic SequencesExample Find "d" of the arithmetic sequence with a15 = 42 and a1=3.
Example Find the term number n of the arithmetic sequence with an = 6, a1=34 and d = 4.
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9 Find a11 when a1 = 13 and d = 6.
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10 Find a17 when a1 = 12 and d = 0.5
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11 Find a17 for the sequence 2, 4.5, 7, 9.5, ...
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12 Find the common difference d when a1 = 12 and a14 = 6.
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13 Find n such a1 = 12 , an= 20, and d = 2.
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Arithmetic Sequences
Find the missing terms
2, ___ , ___ , 23
4, ___ , ___ , 14
7, ___ , ___, ___, 39
9, ___ , ___ , ___, ___ , 34
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14 Find the missing term: 4, ___ , 16
A 20
B 10
C 6
D 2
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15 Find the missing terms: 10, ___ , ___, 8
A 6, 2
B 6, 2
C 5, 1
D 4, 2
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16 Find the missing terms: 12, ___ , ___, 75
A 27, 51
B 33, 54
C 37, 51
D 34, 58
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17 Find d for the arithmetic sequence: 5, ___ , ___ , ___ , 22