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11.2 – Arithmetic Series Sum of Arithmetic Sequences

Sum of Arithmetic Sequences. Definitions Sequence Series

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Sum of an Arithmetic Series

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Page 1: Sum of Arithmetic Sequences. Definitions Sequence Series

11.2 – Arithmetic SeriesSum of Arithmetic Sequences

Page 2: Sum of Arithmetic Sequences. Definitions Sequence Series

DefinitionsSequence

Series

30,26,22,18ia

30262218

Page 3: Sum of Arithmetic Sequences. Definitions Sequence Series

Sum of an Arithmetic SeriesThe sum of the first terms of an arithmetic

series is given by

2)( 1 n

naanS

Page 4: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

Page 5: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n

Page 6: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n11 a

Page 7: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n11 a100100 a

Page 8: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n11 a100100 a

2)1001(100

100

S

Page 9: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n11 a100100 a

2)1001(100

100

S

)101(50100 S

Page 10: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Sum of an Arithmetic SeriesSum of the first 100 positive integers.

2)( 1 n

naanS

100n11 a100100 a

2)1001(100

100

S

)101(50100 S

5050100 S

Page 11: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

Page 12: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

Page 13: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

Page 14: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

Page 15: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

Page 16: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

Page 17: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

daa )113(113

Page 18: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

daa )113(113

d129105

Page 19: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

daa )113(113

d129105

d1296

Page 20: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

daa )113(113

d129105

d1296d8

Page 21: Sum of Arithmetic Sequences. Definitions Sequence Series

Example – Find the first three termsWhat are the first three terms of the

arithmetic series in which

2)( 1 n

naanS

dnaan )1(1

2)1059(741

n

2)114(741 n

n57741

n13

daa )113(113

d129105

d1296d825,17,9

Page 22: Sum of Arithmetic Sequences. Definitions Sequence Series

Additional ExamplesFind the sum of the first 20 even numbers,

beginning with 2.

What are the first four terms of an arithmetic series in which

Page 23: Sum of Arithmetic Sequences. Definitions Sequence Series

Additional ExamplesFind the sum of the first 20 even numbers,

beginning with 2.

What are the first four terms of an arithmetic series in which

420nS

23,20,17,14

Page 24: Sum of Arithmetic Sequences. Definitions Sequence Series

Assignment

Page 25: Sum of Arithmetic Sequences. Definitions Sequence Series

Assignment

State whether each statement is true or false. Explain.