9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2...
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Transcript of 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2...
![Page 1: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/1.jpg)
9-4 Sequences & Series
![Page 2: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/2.jpg)
Basic Sequences
Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2k, … {1/k: k = 1, 2, 3, …} (a1, a2, a3, …, ak, …}
Finite Sequence: has a definite end / last term Infinite Sequence: continues infinitely
![Page 3: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/3.jpg)
Explicit vs. Recursive
• Explicit formula: A function used to find the required term.
• Recursive formula: A function that uses the previous terms to find the required term.
![Page 4: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/4.jpg)
Explicit Sequence
• Ex: Find the first 6 terms and the 100th term of the explicitly-defined sequence
cn = n3 – n
c1
c2
c3
c4
c5
c6
c100
![Page 5: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/5.jpg)
Recursive Sequence
• Ex: Find the first 4 terms and the 8th term of the recursively-defined sequence
a1 = 8 and an = an-1 – 4, for n ≥ 2
a1
a2
a3
a4
a8
![Page 6: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/6.jpg)
Arithmetic SequenceThe pattern is addition!
• A sequence {an} is an arithmetic sequence if it can be written explicitly in the form
an = a1 + (n – 1)d
for some constant d, where d is the common difference (aka pattern number)
• Each term can be obtained recursively by
an = an-1 + d (for all n ≥ 2)
![Page 7: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/7.jpg)
Arithmetic Sequence Example
• Ex: For the arithmetic sequence below, finda) The common difference
b) The tenth term
c) A recursive rule for the nth term
d) An explicit rule for the nth term
6, 10, 14, 18, …
![Page 8: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/8.jpg)
You try!
• Ex: For the arithmetic sequence below, finda) The common difference
b) The tenth term
c) A recursive rule for the nth term
d) An explicit rule for the nth term
4, 1, -2, -5, …
![Page 9: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/9.jpg)
Geometric SequenceThe pattern is multiplication!
• A sequence {an} is a geometric sequence if it can be written explicitly in the form
an = a1 · r n – 1
for some nonzero constant r, where r is the common ratio (aka pattern number)
• Each term can be obtained recursively by
an = an-1 · r (for all n ≥ 2)
![Page 10: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/10.jpg)
Geometric Sequence Example
• Ex: For the geometric sequence below, finda) The common ratio
b) The tenth term
c) A recursive rule for the nth term
d) An explicit rule for the nth term
2, 6, 18, 54, …
![Page 11: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/11.jpg)
You try!
• Ex: For the geometric sequence below, finda) The common ratio
b) The tenth term
c) A recursive rule for the nth term
d) An explicit rule for the nth term
1, -2, 4, -8, 16, …
![Page 12: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/12.jpg)
Constructing Sequences
• Ex: The second and fifth terms of a sequence are 6 and 48, respectively. Find explicit and recursive formulas for the sequence if it is a) arithmetic and b) geometric.
![Page 13: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/13.jpg)
Fibonacci Sequence
• A famous example of a recursive sequence• a1=0, a2=1, an = an-1 + an-2 for n ≥ 3
• 0, 1, 1, 2, 3, 5, 8, 13, 21, …• Named for Leonardo of Pisa• Appears everywhere in nature (check
phyllotaxy in Biology)• Be amazed by Fibonacci
Super Doodle Girl!• http://youtube.com/v/P0tLbl5LrJ8
![Page 14: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/14.jpg)
It’s a race!
• Who can be the first one to find the sum of all numbers from 1 – 100 ?
![Page 15: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/15.jpg)
Sigma Notation• This is a shorthand way to represent a large
sum of numbers• Uses the capital Greek letter sigma, Σ• In summation notation, the sum of the terms of
the sequence {a1, a2, …, an} is denoted
which is read “the sum of ak from k=1 to n”
• The variable k is called the index of summation
![Page 16: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/16.jpg)
…Say what?!!??
• See if you can determine the number represented by each of the following expressions:
1. 2. 3.
![Page 17: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/17.jpg)
Sum of a Finite Arithmetic Sequence
• Let {a1, a2, a3, …, an} be a finite arithmetic sequence with common difference d. Then the sum of the terms of the sequence is
• Proof is on pg 740 if you’re in the mood for some fun!
![Page 18: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/18.jpg)
Revisit Arithmetic Sequences• Remember our example 3, 6, 9, 12, 15?
Find the sum for this sequence. Use the formula.
• What about the sum of numbers 1 – 100?
![Page 19: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/19.jpg)
Sum of a Finite Geometric Sequence
• Let {a1, a2, a3, …, an} be a finite geometric sequence with common ratio r ≠1. Then the sum of the terms of the sequence is
S =
• Proof is on pg 742 if you want more fun!
![Page 20: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/20.jpg)
Revisit Geometric Sequences• Remember our example 2, 4, 8, 16, 32?
Find the sum for this sequence. Use the formula.
• Find the sum for 42, 7, , …,
![Page 21: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/21.jpg)
Infinite Series:• Used when adding an infinite number of
terms together• Not a true sum; how can you find an answer
for infinity?• We use a sequence of partial sums and
limits to find these infinite sums• We can only find the sums if the series
converges to a single value. If it diverges, the limit DNE and we have no sum.
![Page 22: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/22.jpg)
Does it converge?
• For each of the following series, find the first five terms in the sequence of partial sums. Which of the series appear to converge?
1. 0.3 + 0.03 + 0.003 + 0.0003 + …
2. 1 – 2 + 3 – 4 + 5 – 6 + …
![Page 23: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/23.jpg)
Sum of an Infinite Geometric Series
• The geometric series converges
if and only if |r| < 1. If it does converge,
the sum is S =
• Try this formula with #1 from the last slide!
![Page 24: 9-4 Sequences & Series. Basic Sequences Observe patterns! 3, 6, 9, 12, 15 2, 4, 8, 16, 32, …, 2 k, … {1/k: k = 1, 2, 3, …} (a 1, a 2, a 3, …,](https://reader030.fdocuments.in/reader030/viewer/2022032800/56649d305503460f94a09852/html5/thumbnails/24.jpg)
One more neat trick…
• Ex: Express the repeating decimal 7.1414141414 in fraction form.