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ARITHMETIC SEQUENCES AND SERIES, Schemes and Mind Maps of Calculus

Means the sum of the terms in a sequence. Since sequences are infinite, the sequence to be summed must have a specified beginning term and ending term. Explicit ...

Typology: Schemes and Mind Maps

2021/2022

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ARITHMETICSEQUENCESANDSERIES
Asequenceisanorderedlistofnumbers.Eachnumberinthelist
iscalledatermofthesequence.
Ex1a.4,7,10,13,󰇛plus 3󰇜
fifthterm16
b.4,2,1,
,…󰇡times
󰇢
fifthterm
c.1,4,9,16,(squares)
fifthterm25
SequenceNotation
Thefirsttermofasequenceis𝒂𝟏
Thesecondtermofasequenceis𝒂𝟐
Thethirdtermofasequenceis𝒂𝟑
Theterminthenthpositionofasequenceis𝒂𝒏
Thetermpreceding𝑎is𝒂𝒏𝟏

pf3
pf4
pf5
pf8

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ARITHMETIC SEQUENCES AND SERIES

A sequence is an ordered list of numbers. Each number in the list is called a term of the sequence.

Ex1 a. 4, 7, 10, 13, … ሺ plus 3ሻ fifth termൌ 16

b. 4, 2, 1, ଵଶ , … ቀ times ଵଶቁ

fifth termൌ ଵସ

c. 1, 4, 9, 16, … (squares) fifth termൌ 25

Sequence Notation

The first term of a sequence is 𝒂𝟏

The second term of a sequence is 𝒂𝟐

The third term of a sequence is 𝒂𝟑

The term in the n th^ position of a sequence is 𝒂𝒏

The term preceding 𝑎௡ is 𝒂𝒏ି𝟏

Recursive Sequence

A sequence is defined recursively if the first term is given ሺ𝑎ଵ ሻ and there is a formula for determining the n th^ term ሺ𝑎௡ ሻ^ by using the preceding term ሺ𝑎௡ିଵ ሻ.

Ex2 a. List the first 5 terms of the sequence.

𝑎ଵ ൌ 9 and 𝑎௡ ൌ ଵଷ ∗ 𝑎௡ିଵ

Means the sum of the explicit sequence 5𝑘 െ 7 beginning with the 3 rd^ term and ending with the 7th^ term in the sequence.

Means the sum of the terms in a sequence. Since sequences are infinite, the sequence to be summed must have a specified beginning term and ending term.

Explicit Sequence

A sequence is defined explicitly if there is a formula for determining the n th^ term ሺ𝑎௡ ሻ^ independent from the preceding term.

Ex3 Find the first 3 terms and the 100 th^ term of the sequence. 𝑎௡ ൌ 𝑛ሺ3𝑛 ൅ 4ሻ

𝑎ଵ ൌ 1ሺ3ሺ1ሻ ൅ 4ሻ^ 𝑎ଷ ൌ 3ሺ3ሺ3ሻ ൅ 4ሻ 𝑎ଵ ൌ 1ሺ3 ൅ 4ሻ 𝑎ଷ ൌ 3ሺ9 ൅ 4ሻ 𝑎ଵ ൌ 1ሺ7ሻ ൌ 7 𝑎ଷ ൌ 3ሺ13ሻ ൌ 39 𝑎ଶ ൌ 2ሺ3ሺ2ሻ ൅ 4ሻ^ 𝑎ଵ଴଴ ൌ 100ሺ3ሺ100ሻ ൅ 4ሻ 𝑎ଶ ൌ 2ሺ6 ൅ 4ሻ 𝑎ଵ଴଴ ൌ 100ሺ300 ൅ 4ሻ 𝑎ଶ ൌ 2ሺ10ሻ ൌ 20 𝑎ଵ଴଴ ൌ 100ሺ304ሻ ൌ 30, ሼ7, 20, 39, … ሽ 𝑎ଵ଴଴ ൌ 30,

Sigma Notation

௞ୀଷ

෍ሺ3𝑘 ൅ 2ሻ

௞ୀଷ

expanded form Ex ൌ ሺ3 ∗ 3 ൅ 2ሻ ൅ ሺ3 ∗ 4 ൅ 2ሻ ൅ ሺ3 ∗ 5 ൅ 2ሻ ൅ሺ3 ∗ 6 ൅ 2ሻ ൅ ሺ3 ∗ 7 ൅ 2ሻ ൌ 11 ൅ 14 ൅ 17 ൅ 20 ൅ 23 ൌ 85

Arithmetic Sequence

An arithmetic sequence is a sequence in which there is a common difference ሺ𝒅ሻ between consecutive terms.

 Recursive form of an arithmetic sequence 𝑎ଵ ൌ a beginning number, and 𝑎௡ ൌ 𝑎௡ିଵ ൅ 𝑑 (you must know 𝑎ଵ , 𝑎௡ିଵ and 𝑑)  Explicit form of an arithmetic sequence 𝑎௡ ൌ 𝑎ଵ ൅ ሺ𝑛 െ 1ሻ ∗ 𝑑 (you must know 𝑎ଵ and 𝑑)

Ex5 a. Find the 21 st^ term of this arithmetic sequence 3.7, 3.3, 2.9, … From this information we know 𝑎ଵ ൌ 3.7 and 𝑑 ൌ െ. 𝑎ଶଵ ൌ 3.7 ൅ ሺ21 െ 1ሻ ∗ െ. 𝑎ଶଵ ൌ 3.7 ൅ ሺ20 ∗ െ.4ሻ 𝑎ଶଵ ൌ 3.7 ൅ ሺെ8ሻ 𝑎ଶଵ ൌ െ4.

௡ୀଵ

Arithmetic Series

The sum of an arithmetic sequence is known as an arithmetic series. 1+2+3+4+...+97+98+99+

Remember, an arithmetic sequence is infinite, therefore only a partial sum of an arithmetic sequence (finite series) can be computed.

If ሼ𝑎௡ ሽ^ is an arithmetic sequence, and 𝑘 is counting number, then:

ൌ ௞ଶ ሺ𝑎ଵ ൅ 𝑎௞ ሻ^ or 𝑘 ቀ௔^ భ^ ା௔ଶ ೖቁ

Ex7 a. Find the indicated sum of the arithmetic sequence

ଵ଴ if^ 𝑎ଵ^ ൌ െ4^ and^ 𝑑 ൌ^

ଵ ଶ

ଵ଴ ൌ^

ଵ଴ ଶ ሺെ4 ൅ 𝑎ଵ଴^ ሻ^ 𝑎ଵ଴^ ൌ െ4 ൅ ሺ10 െ 1ሻ ∗^

ଵ ଶ 𝑎ଵ଴ ൌ െ4 ൅ ቀ9 ∗ ଵଶ ቁ 𝑎ଵ଴ ൌ െ4 ൅ ଽଶ 𝑎ଵ଴ ൌ ଵଶ

ଵ଴ ൌ 5 ቀെ4 ൅^

ଵ ଶ ቁ

ଵ଴ ൌ 5 ቀെ^

଻ ଶ ቁ ଵ଴ ൌ െ^

ଷହ ଶ

෍ሺ3𝑛 ൅ 7ሻ

ଶ଴

௡ୀଵ

b. Find the indicated sum of the arithmetic sequence

ଶ଴ ൌ^

ଶ଴ ଶ ሺ𝑎ଵ^ ൅ 𝑎ଶ଴^ ሻ^ 𝑎ଵ^ ൌ ሺ3 ∗ 1ሻ ൅ 7 𝑎ଵ ൌ 3 ൅ 7 ൌ 10

𝑎ଶ଴ ൌ ሺ3 ∗ 20ሻ ൅ 7 𝑎ଶ଴ ൌ 60 ൅ 7 ൌ 67 ଶ଴ ൌ 10ሺ10 ൅ 67ሻ

ଶ଴ ൌ 10ሺ77ሻ ଶ଴ ൌ 770

c. Find the indicated sum of the arithmetic sequence

ଷଵ for^ 3.5 ൅ 4 ൅ 4.5 ൅ ⋯ 𝑎ଵ ൌ 3.5 and 𝑑 ൌ.

ଷଵ ൌ^

ଷଵ ଶ ሺ3.5 ൅ 𝑎ଷଵ^ ሻ^ 𝑎ଷଵ^ ൌ 3.5 ൅ ሺ31 െ 1ሻ ∗. 𝑎ଷଵ ൌ 3.5 ൅ ሺ30 ∗ .5ሻ 𝑎ଷଵ ൌ 3.5 ൅ 15 𝑎ଷଵ ൌ 18. ଷଵ ൌ^

ଷଵ ଶ ሺ3.5 ൅ 18.5ሻ

ଷଵ ൌ^

ଷଵ ଶ ሺ22ሻ ଷଵ ൌ^

଺଼ଶ ଶ ଷଵ ൌ 341