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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
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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 ሺ𝑎ିଵ ሻ.
Ex2 a. List the first 5 terms of the sequence.
𝑎ଵ ൌ 9 and 𝑎 ൌ ଵଷ ∗ 𝑎ିଵ
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 ൌ 𝑎ିଵ 𝑑 (you must know 𝑎ଵ , 𝑎ିଵ 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 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
