Download Lesson 12.7: Sequences and Series Sequences 12.7 ­ Sequences and Series.notebook

12.7 ­ Sequences and Series.notebook
June 07, 2013
Lesson 12.7:
Sequences and Series
May 30­7:11 AM
Sequences
Definition: A sequence is a set of numbers in a specific order. 2, 5, 8,…. is an example of a sequence.
Note: A sequence may have either a finite or an infinite number of terms.
The terms of a sequence are the individual numbers in the sequence. If we let a1 represent the first term of a sequence, an represent the nth term, and n represent the term number, then the sequence is represented by a1, a2, a3, …, a n, … In the example above, a1=2, a2=5, a3= 8, etc.
Example: Write the first 3 terms of each sequence.
1. an = n2 + 2
2. an = (n+1)(n+2)
a1 = 1 + 2 = 3
a2 = 22 + 2 = 6
a3 = 32 + 2 = 11
2
3. an = 2an­1 + 3, a1 = 5
Mar 29­9:45 AM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
arithmetic sequences
Definition: An arithmetic sequence is a sequence in which each term, after the first, is found by adding a constant number to the previous term. That constant number is called the common difference.
Example: In the sequence 2, 5, 8, 11, 14, ...... This is an arithmetic sequence because to get each number in the sequence, we add 3. So, we say the common difference is 3.
Practice: Determine if the sequences below are arithmetic sequences. If yes, determine the common difference.
1.) ­2, 3, 8, …..
2.) 5, 9, 13, …..
3.) 0, 7, 14, ……
4.) 5, ­1, ­7, …….
5.) ­9x, ­2x, 5x, …….
Mar 29­9:50 AM
How to write an arithmetic sequence:
Example: Write the first five terms of the arithmetic sequence in which and d are given a1 =17, d = 12
Start with a1 and add the common difference (d) to the term. Continue adding to find more terms
a1 Practice: Write the first five terms of the arithmetic sequence in which and d are given as follows
1.
a1 = 19, d = 6
2. a1 = 27, d = ­4
3. a1 = 3, d = Mar 29­11:44 AM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
How to find any term of an arithmetic sequence
th
If a1 is the first term of an arithmetic sequence, a
n the n term, d is the common difference, a formula for finding the value of the nth term of an arithmetic sequence is:
an = a1 + (n ­ 1)d
Example: Find the 75th term of the sequence 2, 5, 8,……
1. Find the 13th term of 2, 8, 14, 20, 26, …..
2. Find the 43rd term of ­19, ­15, ­11, …..
th term is 17.
3. Write the arithmetic sequence whose first term is 5 and whose 7
st term is 4 and 4. Find the common difference (d) in the arithmetic sequence whose 1
whose 11th term is 64.
rd term 5. Find the common difference in the arithmetic sequence whose 3
is 0 and whose 29th term is 50.
Mar 29­11:49 AM
Geometric Sequences
Definition: A geometric sequence is a sequence in which each term, after the first is formed by multiplying the previous term by a constant number, which is called the common ratio.
For example: 3, 6, 12, 24, … is a geometric sequence, each term is being multiplied by 2 to produce the next term. ­ a1 represents the first term (3),
­ an the nth term, ­ r is the common ratio (multiply by 2, so r = 2) .
To find r (the common ratio): divide any term in the sequence by the one before it.
Example: Determine if the sequence is geometric. If it is, give the common ratio and the next two terms.
1. 2, ­6, 18, ­54, 162, …. 2. 5, 10, 15, 20, 25, .....
3. 10, 30, 90, 270, 810, .....
4. 64, 32, 16, 8, 4, ......
Example:
Write the first four terms of the geometric sequence in which a1 and r are as follows.
5.
= 13, r = 7
6. a1 = 4, r = 7.
=7, r = ­2
Mar 29­12:10 PM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
How to find any term of a geometric sequence
If a1 represents the first term of a geometric sequence, a n the n th term, and r the common ratio, then the formula for finding the n th term is:
an = a1 rn ­ 1
Example:
1) Find the 8th term of the sequence 243, 81, 27, 9, ….
Practice: Find the term indicated in each of the following geometric sequence.
2. 10th term of 3, 6, 12, ……..
3. 6th term of , ­1, 2, ­4, …
Mar 30­7:51 AM
How to find r (the common ratio)
Use the same formula: a n = a1rn­1
1. Find the common ratio of the geometric sequence, whose first term is 5 and whose 4th term is ­320.
2. Find the common ratio of the geometric sequence whose first term is 4 and whose 3rd term is 36.
Mar 30­7:58 AM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
Series
The sum of the terms of a sequence is called a series. Mar 28­5:39 PM
Arithmetic Series
To find the sum of a certain number of terms of an arithmetic sequence:
Sn = n(a1 + an)
2
where Sn is the sum of n terms (nth partial sum),
a1 is the first term, an is the nth term.
To find an, you may have to use the formula for finding a term in an arithmetic sequence:
an = a1 + (n ­ 1)d
Mar 28­5:44 PM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
Example
Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...
a1 = 4
n = 20
d = 2
Step 2:
To use the sum formula, a n needs to be found first. an = a1 + (n ­ 1)d
ANSWER
Step 1:
a20 = 4 + (20 ­ 1)(2)
a20 = 4 + 19(2)
a­20 = 4 + 38
a20 = 42
Now, we can use the sum formula to solve.
460
Sn = n(a1 + an)
2
Mar 28­5:52 PM
Practice
1. Find the sum of the first 30 terms of:
5, 9, 13, 17, . . . 2. Determine the sum of the first 17 terms of the arithmetic sequence whose first 4 terms are: ­15, ­9, ­3, 3
3. Determine the sum of the first 8 terms of the arithmetic sequence whose first 4 terms are 8, 11, 14, 17
Mar 28­6:04 PM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
4. Find the sum of the arithmetic series 3, 6, 9, .... ,99
Hint:
Use the sum formula, n needs to be found first. 5. Determine the sum of 22, 16, 10, … , ­80
Mar 28­6:08 PM
Geometric Series
To find the sum of a certain number of terms of a geometric sequence:
Sn = a1(1 ­ rn)
1 ­ r
where Sn is the sum of n terms (nth partial sum), a1 is the first term, r is the common ratio.
Mar 28­6:17 PM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
Example
Find the sum of the first 8 terms of the sequence ­5, 15, ­45, 135, ... a1 = ­5
r = ­3
n = 8
Sn = a1(1 ­ rn)
1 ­ r
8200
Mar 28­6:24 PM
practıce
1) Determine the sum of the first 15 terms of the geometric sequence 1, 2, 4, 8, …
2) Determine the sum of the first 11 terms of the geometric sequence 2, ­6, 18, ­54, …
3) Determine the sum of the first 6 terms of the geometric sequence 1000, 200, 40, 8, …
4) Determine the sum of the first 9 terms of the geometric sequence 1, 6, 36, 216, …
Mar 28­6:28 PM
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12.7 ­ Sequences and Series.notebook
June 07, 2013
Writing Series in Sigma Notation
To write a series using sigma notation, look for patterns! See if the series is arithmetic or geometric.
If it is arithmetic, use the form:
here, d = common difference. Adjust c and r to fit the series you need. If it is geometric, use the form:
here, r = common ratio. Again, adjust k and r accordingly.
May 30­7:55 AM
1) Use sigma notation to represent the sum of the first 30 terms of the series:
4 + 8 + 12 + 16 + ...
2) Use sigma notation to represent the sum of the first 15 terms of the series:
­1 ­ 3 ­ 5 ­ 7 ­ 9 ­ ...
3) Use sigma notation to represent the sum of the first 25 terms of the series:
2­1 + 2­2 + 2­3 + 2­4 + ...
4) Use sigma notation to represent the sum of the first 40 terms of the series:
6 + 18 + 54 + 162 + ...
May 30­8:03 AM
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