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1. Sequences and Series
Mathematics 4
February 1, 2012
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2. Infinite Sequences
An infinite sequence (or simply sequence) is a function whose domain
is the set of positive integers.
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3. Infinite Sequences
An infinite sequence (or simply sequence) is a function whose domain
is the set of positive integers.
Example: an = 3n + 1
term number 1 2 3 4 5 n
term value 4 7 10 13 16 3n + 1
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4. Infinite Sequences
An infinite sequence (or simply sequence) is a function whose domain
is the set of positive integers.
Example: an = 3n + 1
term number 1 2 3 4 5 n
term value 4 7 10 13 16 3n + 1
This sequence is said to be explicitly defined.
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5. Explicitly Defined Infinite Sequences
List the first three terms and the tenth term of each sequence
n
1. an =
n+1
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6. Explicitly Defined Infinite Sequences
List the first three terms and the tenth term of each sequence
n
1. an =
n+1
1 2 3 10
2 , 3 , 4 , a10 = 11
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7. Explicitly Defined Infinite Sequences
List the first three terms and the tenth term of each sequence
n
1. an =
n+1
1 2 3 10
2 , 3 , 4 , a10 = 11
n2
2. an = (−1)n+1
3n − 1
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8. Explicitly Defined Infinite Sequences
List the first three terms and the tenth term of each sequence
n
1. an =
n+1
1 2 3 10
2 , 3 , 4 , a10 = 11
n2
2. an = (−1)n+1
3n − 1
1
2. − 4 , 8 , a10 = − 100
5
9
29
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9. Recursively Defined Infinite Sequences
A recursively defined sequence is a sequence where the first term a1 is
stated, together with a rule for obtaining the next term an+1 from the
preceding term an .
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10. Recursively Defined Infinite Sequences
A recursively defined sequence is a sequence where the first term a1 is
stated, together with a rule for obtaining the next term an+1 from the
preceding term an .
List the first five terms of each sequence
1. a1 = 1, an+1 = 7 − 2an
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11. Recursively Defined Infinite Sequences
A recursively defined sequence is a sequence where the first term a1 is
stated, together with a rule for obtaining the next term an+1 from the
preceding term an .
List the first five terms of each sequence
1. a1 = 1, an+1 = 7 − 2an
1, 5, −3, 13, −19, 45
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12. Recursively Defined Infinite Sequences
A recursively defined sequence is a sequence where the first term a1 is
stated, together with a rule for obtaining the next term an+1 from the
preceding term an .
List the first five terms of each sequence
1. a1 = 1, an+1 = 7 − 2an
1, 5, −3, 13, −19, 45
2. a1 = a2 = 1, an = an−1 + an−2
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13. Recursively Defined Infinite Sequences
A recursively defined sequence is a sequence where the first term a1 is
stated, together with a rule for obtaining the next term an+1 from the
preceding term an .
List the first five terms of each sequence
1. a1 = 1, an+1 = 7 − 2an
1, 5, −3, 13, −19, 45
2. a1 = a2 = 1, an = an−1 + an−2
1, 1, 2, 3, 5
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14. Infinite Sequences
Examples
1. The number of bacteria in a certain culture is initially 200, and the
culture doubles in size every hour. Find an explicit and a recursive
formula for the number of bacteria present after n hours.
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15. Infinite Sequences
Examples
1. The number of bacteria in a certain culture is initially 200, and the
culture doubles in size every hour. Find an explicit and a recursive
formula for the number of bacteria present after n hours.
2. Use a calculator to determine the value of a4 in the sequence
a1 = 3, an+1 = an − tan an .
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16. Arithmetic Sequences
A sequence a1 , a2 , a3 , ..., an , ... is an arithmetic sequence if each term
after the first is obtained by adding the same fixed number d to the
preceding term.
an+1 = an + d
The number d = an+1 − an is called the common difference of the
sequence.
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17. Arithmetic Sequences
Given the diagram below:
1. Determine the common difference between diagrams.
2. How many blocks will Diagram 10 have?
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18. Arithmetic Sequences
Finding the nth term of an AS
The nth term of an arithmetic sequence is given by:
an = a1 + (n − 1)d
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19. Arithmetic Sequences
an = a1 + (n − 1)d
1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...
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20. Arithmetic Sequences
an = a1 + (n − 1)d
1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...
2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Find
the 25th term.
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21. Arithmetic Sequences
an = a1 + (n − 1)d
1. Find the 25th term of the arithmetic sequence 2, 5, 8, 11, ...
2. The 18th and 52nd terms of an AS are 3 and 173 respectively. Find
the 25th term.
3. The terms between any two terms of an arithmetic sequence are
called the arithmetic means between these two terms. Insert four
arithmetic means between -1 and 14.
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22. Arithmetic Sequences
Partial Sum of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence is given by the
formula:
n[2a1 + (n − 1)d]
Sn =
2
or
n(a1 + an )
Sn =
2
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23. Arithmetic Sequences
n(a1 + an ) n[2a1 + (n − 1)d]
Sn = =
2 2
1. Find the sum of the first 30 terms of the arithmetic sequence -15,
-13, -11, ...
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24. Arithmetic Sequences
n(a1 + an ) n[2a1 + (n − 1)d]
Sn = =
2 2
1. Find the sum of the first 30 terms of the arithmetic sequence -15,
-13, -11, ...
2. The sum of the first 15 terms of an arithmetic sequence is 270.
Find a1 and d if a15 = 39.
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25. Harmonic Sequences
A harmonic sequence is a sequence of numbers whose reciprocals
form an arithmetic sequence.
1. Find two harmonic means between 4 and 8.
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26. Harmonic Sequences
A harmonic sequence is a sequence of numbers whose reciprocals
form an arithmetic sequence.
1. Find two harmonic means between 4 and 8.
2. Find the 14th term of the harmonic sequence starting with 3, 1.
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27. Homework 6
Vance p. 179 numbers 2, 4, 6, 10, 12, 14, 15, 18, 24, 25.
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28. Geometric Sequences
A geometric sequence is a sequence in which each term after the first
is obtained by multiplying the same fixed number, called the common
ratio, by the preceding term.
gn+1 = gn · r
gn+1
The number r = is called the common ratio of the sequence.
gn
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29. Geometric Sequences
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30. Geometric Sequences
gn+1 = gn · r
Examples:
1. Give the next 3 terms of the GS 27, 9, 3, ...
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31. Geometric Sequences
gn+1 = gn · r
Examples:
1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1
3 9
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32. Geometric Sequences
gn+1 = gn · r
Examples:
1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1
3 9
2. Find the 10th term of the GS -8, 4, -2, ...
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33. Geometric Sequences
gn+1 = gn · r
Examples:
1. Give the next 3 terms of the GS 27, 9, 3, ... 1, 1 , 1
3 9
1
2. Find the 10th term of the GS -8, 4, -2, ... 64
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34. Geometric Sequences
gn = g1 · rn−1
A geometric sequence can also be expressed explicitly as:
gn = g1 · rn−1
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35. Geometric Sequences
gn = g1 · rn−1
A geometric sequence can also be expressed explicitly as:
gn = g1 · rn−1
Examples
1. If the 8th term of a GS is 243 and the 5th term is 9, write the first
3 terms.
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36. Geometric Sequences
gn = g1 · rn−1
A geometric sequence can also be expressed explicitly as:
gn = g1 · rn−1
Examples
1. If the 8th term of a GS is 243 and the 5th term is 9, write the first
1 1
3 terms. 9, 3, 1
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37. Geometric Sequences
gn = g1 · rn−1
A geometric sequence can also be expressed explicitly as:
gn = g1 · rn−1
Examples
1. If the 8th term of a GS is 243 and the 5th term is 9, write the first
1 1
3 terms. 9, 3, 1
2. Find the 1st term of a GS with g5 = 162 and r = −3.
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38. Geometric Sequences
gn = g1 · rn−1
A geometric sequence can also be expressed explicitly as:
gn = g1 · rn−1
Examples
1. If the 8th term of a GS is 243 and the 5th term is 9, write the first
1 1
3 terms. 9, 3, 1
2. Find the 1st term of a GS with g5 = 162 and r = −3. 2
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39. Sum of a Geometric Sequence
The sum of n terms of any geometric sequence is given by the
formula:
g1 (1 − rn )
Sn = , r=1
1−r
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40. Geometric Sequences
Exercises
1. Find the value of k so that 2k + 2,5k − 11, and 7k − 13 will form a
geometric sequence.
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2. Insert four geometric means between 4 and 125 .
3. A man accepts a position at P360,000 a year with the
understanding that he will receive a 2% increase every year. What
will his salary be after 10 years of service?
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41. Homework 7
Vance p. 311 numbers 3, 4, 9, 11, 14, 15, 17, 18, 22, 23
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