8.1 sequences comp.notebook - Amphitheater Public Schools...Each term in an arithmetic sequence can...
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8.1 sequences comp.notebook 1 January 08, 2020 Dec 273:54 PM 81 Sequences Defining a Sequence A sequence {a n } is a list of numbers written in explicit order. e.g. If the domain is finite, then the sequence is a Ex.1 Explicit Sequence Find the first six terms and the 100 th term of the sequence {a n }, where This example is defined because it is defined in terms of n.
Transcript of 8.1 sequences comp.notebook - Amphitheater Public Schools...Each term in an arithmetic sequence can...
8.1 sequences comp.notebook
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81 Sequences
Defining a Sequence
A sequence {an} is a list of numbers written in explicit order.e.g.
If the domain is finite, then the sequence is a
Ex.1 Explicit Sequence
Find the first six terms and the 100th term of the sequence {an}, where
This example is defined because it is defined in terms of n.
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Recursive Sequences depend on what has gone on before.
Ex. 2Find the first three terms and the sixth term for the recursive sequence defined by the following conditions:
b1=5bn=2bn1 3 for ∀n≥2
TRYRepeat example 2 with the following conditions:
b1= 4b2 = 7bn=3bn2 + 2bn1 for ∀n≥3
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Arithmetic Sequences
Each term in an arithmetic sequence can be obtained recursively from its preceding term by adding d.
d is the
Recursive definition:
Explicit definition:
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Ex. 3 Defining Arithmetic Sequences
For the arithmetic sequence {7,3,1,5,9,...} find:
(a) the common difference(b) a recursive rule for the nth term(c) an explicit rule for the nth term(d) the 42nd term
Ex.4 Repeat the above for the arithmetic sequence {ln2, ln6, ln18, ln54,...}
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Ex. 5 Constructing a Sequence
The third and sixth terms of an arithmetic sequence are 5 and 14, respectively. Find the common difference, first term, and an explicit rule for the nth term.
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Geometric Sequences
Each term in a geometric sequence can be obtained recursively from its preceding term by multiplying by r.
r is the
Recursive definition:
Explicit definition:
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Ex. 6 Defining Geometric Sequences
For the geometric sequence {4,12,36,108,...} find:
(a) the common ratio(b) a recursive rule for the nth term(c) an explicit rule for the nth term(d) the seventh term
Ex. 7Repeat example 6 for the geometric sequence {53,55,57,59,...}
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Ex. 8 Constructing a Sequence
The third and sixth terms of a geometric sequence are 20 and 160, respectively. Find the common ratio, first term, and an explicit rule for the nth term.
TRY
Repeat above for a3 = 25 and a6 = 1/5
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Limit of a Sequence
Some sequences do not have a limit e.g.
Some sequences tend towards a limit as n>∞ e.g.
Notation:
L is the limit of the sequence. If L exists we say the sequence CONVERGES to L.
Sequences that do not have limits DIVERGE.
Properties of Limits
If L and M are real numbers and and , then
1) Sum Rule:
2) Difference Rule:
3) Product Rule:
4) Quotient Rule:
5) Constant Multiple Rule:
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Ex. 9 Finding the Limit of a Sequence
Determine whether the sequence converges or diverges. If it converges, find its limit.
(a) (b)
Ex. 10 Determining Convergence or Divergence
Determine whether the sequence with the given nth term converges or diverges. If it converges, find its limit.
(a) n=1,2,3,... (b) b1=4, bn=bn1+2 ∀n≥2
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Sandwich Theorem for Sequences
If and there is an integer N for which
an≤bn≤cn for ∀n>N, then .
Ex. 11
Show that the sequence converges, and find its limit.
Ex. 12
Determine if the sequence converges, if it converges find its limit.
Factorials
In your groups:
Evaluate 4!
Expand n! 2 different ways.
What is ? What is ?
Which one grows faster?
What is ? What is ?
Which one gets smaller more quickly?
