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Pre-Calculus
Sequences and Series
www.njctl.org
2015-03-24
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Table of Contents
Arithmetic Sequences
Arithmetic Series
Geometric Sequences
Geometric Series
Special Sequences
Infinite Geometric Series
Binomial Theorem
click on the topic to go to that section
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1 Find the next term in the arithmetic sequence: 3, 9, 15, 21, . . .
Arithmetic Sequence
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2 Find the next term in the arithmetic sequence: -8, -4, 0, 4, . . .
Arithmetic Sequence
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3 Find the next term in the arithmetic sequence: 2.3, 4.5, 6.7, 8.9, . . .
Arithmetic Sequence
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4 Find the value of d in the arithmetic sequence: 10, -2, -14, -26, . . .
Arithmetic Sequence
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5 Find the value of d in the arithmetic sequence: -8, 3, 14, 25, . . .
Arithmetic Sequence
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Arithmetic SequenceAs we study sequences we need a way of naming the terms. a1 to represent the first term,a2 to represent the second term,a3 to represent the third term,and so on in this manner.
If we were talking about the 8th term we would use a8.
When we want to talk about general term call it the nth termand use an.
Arithmetic Sequence
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Arithmetic SequenceWrite the first four terms of the arithmetic sequence that is described.
a1 = 4; d = 6
a1 = 3; d = -3
a1 = 0.5; d = 2.3
a2 = 7; d = 5
Arithmetic Sequence
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6 Which sequence matches the description?
A 4, 6, 8, 10
B 2, 6,10, 14
C 2, 8, 32, 128
D 4, 8, 16, 32
Arithmetic Sequence
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7 Which sequence matches the description?
A -3, -7, -10, -14
B -4, -7, -11, -13
C -3, -7, -11, -15
D -3, 1, 5, 9
Arithmetic Sequence
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8 Which sequence matches the description?
A 7, 10, 13, 16
B 4, 7, 10, 13
C 1, 4, 7,10
D 3, 5, 7, 9
Arithmetic Sequence
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Arithmetic Sequence
To find a specific term,say the 5th or a5, you couldwrite out all of the terms.
But what about the 100th term(or a100 )?
We need to find a formula to get there directly without writing out the whole list.
Arithmetic Sequence
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Arithmetic Sequence
Consider: 3, 9, 15, 21, 27, 33, 39,. . .
a1 3
a2 9 = 3+6
a3 15 = 3+12 = 3+2(6)
a4 21 = 3+18 = 3+3(6)
a5 27 = 3+24 = 3+ 4(6)
a6 33 = 3+30 = 3+5(6)
a7 39 = 3+36 = 3+6(6)
Do you see a pattern that relates the term number to its
value?
Arithmetic Sequence
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Arithmetic SequenceExample Find the 21st term of the arithmetic sequence with a1 = 4 and d = 3.
Example Find the 12th term of the arithmetic sequence with a1 = 6 and d = -5.
Arithmetic Sequence
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Arithmetic SequenceExample Find the 1st term of the arithmetic sequence with a15 = 30 and d = 7.
Example Find the 1st term of the arithmetic sequence with a17 = 4 and d = -2.
Arithmetic Sequence
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Arithmetic SequenceExample Find d of the arithmetic sequence with a15 = 42 and a1=3.
Example Find the term number n of the arithmetic sequence with an = 6, a1=-34 and d = 4.
Arithmetic Sequence
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9 Find a11 when a1 = 13 and d = 6.
Arithmetic Sequence
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10 Find a17 when a1 = 12 and d = -0.5
Arithmetic Sequence
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11 Find a17 for the sequence 2, 4.5, 7, 9.5, ...
Arithmetic Sequence
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12 Find the common difference d when a1 = 12 and a13= 6.
Arithmetic Sequence
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13 Find n such a1 = 12 , an= -20, and d = -2.
Arithmetic Sequence
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14 Tom works at a car dealership selling cars. He is paid $4000 a month plus a $300 commission for every car he sells after the first car . How much did he make in April if he sold 14 cars?
Arithmetic Sequence
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Arithmetic Sequence
Find the missing terms in the arithmetic sequence.
4, 6, 8, 10,___
5, 10, ___, 20
___, 12, 9, 6
6,___, 14
Notice in the last example d was added to get from 6 to ___ and another d was added to get from ___ to 14.
Or 6 + 2d = 14
Arithmetic Sequence
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Arithmetic Sequence
Find the missing terms
2, ___ , ___ , 23
4, ___ , ___ , -14
7, ___ , ___, ___, 39
-9, ___ , ___ , ___, ___ , -34
Arithmetic Sequence
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15 Find the missing term: 4, ___ , -16
A -20
B -10
C -6
D 2
Arithmetic Sequence
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16 Find the missing terms: -10, ___ , ___, 8
A -6, -2
B -6, 2
C -5, 1
D -4, 2
Arithmetic Sequence
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17 Find the missing terms: 12, ___ , ___, 75
A 27, 51
B 33, 54
C 37, 51
D 34, 58
Arithmetic Sequence
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18 Find d for the arithmetic sequence: 5, ___ , ___ , ___ , 21
Arithmetic Sequence
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Arithmetic SeriesAn arithmetic series is the sum of the terms in the arithmetic sequence.
Sn represents the sum of the first n terms.
Consider 4, 7,10, 13, 16, 19, 22, . . .
S4= 4 + 7 + 10 + 13 = 34
S6= 4 + 7 + 10 + 13 + 16 + 19= 69
Arithmetic Series
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19 Consider 3, 9, 15, 21, 27, 33, 39,... what is S4?
Arithmetic Series
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20 Consider 3, 9, 15, 21, 27, 33, 39,... what is S5?
Arithmetic Series
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21 Consider 3, 9, 15, 21, 27, 33, 39,... what is S7?
Arithmetic Series
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Arithmetic SeriesSuppose we wanted to find the first 100 terms of 4, 7,10, 13, 16, 19, 22, . . . or S 100?
There must be a short cut.
The 100th term of the sequence is 301 using a100 = 4 + (100 - 1)3.
S100 = 4 + 7 + 10 +13 + . . . + 292 + 295 +298 + 301If we add the smallest and largest (4 + 301)= 305If we add the next two (7 + 298) = 305and continue (10 + 295) until they are all paired up (151+154).
We now have 50 pairs of 305, so...S100 = 4 + 7 + 10 +. . .+298 + 301 = 50(305) = 15250
Do you see a pattern?
Arithmetic Series
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Find Sn for each arithmetic series.
Example: a1 = 7 and a12 = -23
Example: a1 = 6, n = 10, and d = 9
Arithmetic Series
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Find Sn for each arithmetic series.Example: a12= 30 and d = -7
Example: a1 = 2, an = 32, and d = 5
Arithmetic Series
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22 Find the Sn for the arithmetic series described:a1 = 19 and a12 = 37.
Arithmetic Series
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23 Find the Sn for the arithmetic series described:a1 = 30 and a17 = -45.
Arithmetic Series
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24 Find the Sn for the arithmetic series described:a1 = 20, n = 8, and d = 6.
Arithmetic Series
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25 Find the Sn for the arithmetic series described:an = 20, n = 9, and d = -4.
Arithmetic Series
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26 Find the Sn for the arithmetic series described:17 + 20 + 23 + 26 + 29 + . . . + 50
Arithmetic Series
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Sigma Notation
Sigma ( ) is the Greek letter S.
And means the sum of the terms in a sequence.
The difference between Sn and sigma is that Sn is always the first to the nth term.
means start with 3rd and sum up through the 9th term.
Arithmetic Series
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27 Evaluate
Arithmetic Series
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28 Evaluate
Arithmetic Series
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29 Evaluate
Arithmetic Series
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How would evaluate ?
Arithmetic Series
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30 Evaluate
Arithmetic Series
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31 Find the next term in geometric sequence: 6, -12, 24, -48, 96, . . .
Geometric Sequence
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32 Find the next term in geometric sequence: 64, 16, 4, 1, . . .
Geometric Sequence
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33 Find the next term in geometric sequence: 6, 15, 37.5, 93.75, . . .
Geometric Sequence
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34 Is the following sequence a geomtric one? 48, 24, 12, 8, 4, 2, 1
Yes
No
Geometric Sequence
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Geometric SequenceGeometric Sequences can be described by giving the
first term, a1, and the common ratio, r.
Examples: Find the first five terms of the geometric sequence described.
1) a1 = 6 and r = 3
2) a1 = 8 and r = -.5
3) a1 = -24 and r = 1.5
4) a1 = 12 and r = 2/3
Geometric Sequence
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35 Find the first four terms of the geometric sequence described: a1 = 6 and r = 4.
A 6, 24, 96, 384
B 4, 24, 144, 864
C 6, 10, 14, 18
D 4, 10, 16, 22
Geometric Sequence
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36 Find the first four terms of the geometric sequence described: a1 = 12 and r = -1/2.
A 12, -6, 3, -.75
B 12, -6, 3, -1.5
C 6, -3, 1.5, -.75
D -6, 3, -1.5, .75
Geometric Sequence
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37 Find the first four terms of the geometric sequence described: a1 = 7 and r = -2.
A 14, 28, 56, 112
B -14, 28, -56, 112
C 7, -14, 28, -56
D -7, 14, -28, 56
Geometric Sequence
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Geometric SequenceConsider the sequence: 3, 6, 12, 24, 48, 96, . . .To find the seventh term, just multiply the sixth term by 2.But what if I want to find the 20 th term?
Look for a pattern:a1 3
a2 6 = 3(2)
a3 12 = 3(4) = 3(2)2
a4 24 = 3(8) = 3(2)3
a5 48 = 3(16) = 3(2)4
a6 96 = 3(32) = 3(2)5
a7 192 = 3(64) = 3(2)6
Do you see a pattern?
Geometric Sequence
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Geometric Sequence
Geometric Sequence
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Geometric SequenceFind the indicated term.
Example: a20 given a1 =3 and r = 2.
Example: a10 for 2187, 729, 243, 81
Geometric Sequence
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Geometric SequenceExample: Find r if a6 = .2 and a1 = 625
Example: Find n if a1 = 6, an = 98,304 and r = 4.
Geometric Sequence
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38 Find a12 in a geometric sequence wherea1 = 5 and r = 3.
Geometric Sequence
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39 Find a10 in a geometric sequence wherea1 = 7 and r = -2.
Geometric Sequence
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40 Find a7 in a geometric sequence wherea1 = 10 and r = -1/2.
Geometric Sequence
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41 Find r of a geometric sequence wherea1 = 3 and a10=59049.
Geometric Sequence
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42 Find n of a geometric sequence wherea1 = 72, r = .5, and an = 2.25
Geometric Sequence
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Geometric SequenceFind the missing term in the geometric sequence3, 9, 27,___
5, 1, 1/5 , ____
___, -10, 50, -250
-2, ___, -32
Geometric Sequence
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Geometric Sequence
Find the missing terms in the geometric sequence.
5, ___, ___, 40
-54, ___, ___, 16
4, ___, ___, ___, 324
144, ___, ___, ___, ___, 4.5
Geometric Sequence
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43 What number(s) fill in the blanks of the geometric sequence: ___, 14, 98, 686
A -14
B -7
C -2
D 2
E 7
F 8
G 10
H 12
I 28
J 50
Geometric Sequence
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The sum of a geometric series can be found using the formula:
Examples: Find Sn for each example.
1) a1= 5, r= 3, n= 6
2) a1= -3, r= -2, n=7
Geometric Series
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Sometimes information will be missing, so that using
isn't possible to start. Look to use to find missing information.Example: a1 = 16 and a5 = 243, find S5
Geometric Series
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Geometric Series
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Example: a1 = 16 and a5 = 243, find S5 (continued)
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46 Find the indicated sum of the geometric series described: a1 = 10, n = 6, and r = 6
Geometric Series
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47 Find the indicated sum of the geometric series described: a1 = -2, n = 5, and r = 1/4
Geometric Series
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48 Find the indicated sum of the geometric series described: a1 = 8, n = 6, and r = -2
Geometric Series
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49 Find the indicated sum of the geometric series described: a1 = 8, n = 5, and a6 = 8192
Geometric Series
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50 Find the indicated sum of the geometric series described: r = 6, n = 4, and a4 = 2592
Geometric Series
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51 Find the indicated sum of the geometric series described: 8 - 12 + 18 - . . . find S7
Geometric Series
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Sigma ( )can be used to describe the sum of a geometric series.
Examples:
We can still use , but to do so we must examine the sigma notation.
n = 4 Why? The bounds on below and on top indicate that.
a1 = 6 Why? The coefficient is all that remains when the base is powered by 0.
r = 3 Why? In the exponential chapter this was our growth rate.
Geometric Series
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52 Find the sum:
Geometric Series
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53 Find the sum:
Geometric Series
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54 Find the sum:
Geometric Series
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n Sn
1 3
2 12
3 39
4 120
5 363
6 1,092
7 3,279
8 9,840
n an
1 3
2 9
3 27
4 81
5 243
6 729
7 2,187
8 6,561Because r = 3, the series isgrowing at increasing rate.
Infinite Geometric Series
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n Sn
1 64
2 96
3 112
4 120
5 124
6 126
7 127
8 127.5
n an
1 64
2 32
3 16
4 8
5 4
6 2
7 1
8 0.5 Because r = 1/2 , an --> 0.Notice Sn --> an asymptote?
Infinite Geometric Series
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n Sn
1 64
2 32
3 48
4 40
5 44
6 42
7 43
8 42.5
n an
1 64
2 -32
3 16
4 -8
5 4
6 -2
7 1
8 -0.5 Because r = -1/2 , an --> 0.Notice Sn --> an asymptote?
Infinite Geometric Series
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For an infinite geometric series to approach a value,-1 < r < 1 then
The examples from the previous slides:
Example 1: a1 = 3 and r = 3.
Example 2: a1 = 64 and r= 1/2
Example 3: a1 = 64 and r= -1/2
Infinite Geometric Series
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55 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
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59 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
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60 Find the sum of this infinite geometric series, if one exists:
Infinite Geometric Series
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A recursive formula is one in which to find a term you need to know the preceding term.
So to know term 8 you need the value of term 7, and to know the nth term you need term n-1
In each example, find the first 5 terms
a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
Special Sequences
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61 Find the first four terms of the sequence:
A 6, 3, 0, -3
B 6, -18, 54, -162
C -3, 3, 9, 15
D -3, 18, 108, 648
a1 = 6 and an = an-1 - 3
Special Sequences
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62 Find the first four terms of the sequence:
A 6, 3, 0, -3
B 6, -18, 54, -162
C -3, 3, 9, 15
D -3, 18, 108, 648
a1 = 6 and an = -3an-1
Special Sequences
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63 Find the first four terms of the sequence:
A 6, -22, 70, -216
B 6, -22, 70, -214
C 6, -14, 46, -134
D 6, -14, 46, -142
a1 = 6 and an = -3an-1 + 4
Special Sequences
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a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the first column represents an Arithmetic Sequence.We can write this formula so that we find a n directly.
Recall:
We will need a1 and d,they can be found both from the table and the recursive formula.
Special Sequences
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a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the second column represents a Geometric Sequence.We can write this formula so that we find an directly.
Recall:
We will need a1 and r,they can be found both from the table and the recursive formula.
Special Sequences
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a1 = 6, an = an-1 +7 a1 =10, an = 4an-1 a1 = 12, an = 2an-1 +3
6 a1 10 a1 12
13 a2 40 a2 27
20 a3 160 a3 57
27 a4 640 a4 117
34 a5 2560 a5 237
The recursive formula in the third column represents neither an Arithmetic or Geometric Sequence.
This observation comes from the formula where you have both multiply and add from one term to the next.
Special Sequences
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64 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 12 , an = 2an-1 +7
Special Sequences
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65 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 20 , an = 5an-1
Special Sequences
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66 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
a1 = 20 , an = 5an-1
an = 20 + (n-1)5
an = 20(5)n-1
an = 5 + (n-1)20
an = 5(20)n-1
Special Sequences
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67 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = -12 , an = an-1 - 8
Special Sequences
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68 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = -12 + (n-1)(-8)
an = -12(-8)n-1
an = -8 + (n-1)(-12)
an = -8(-12)n-1
a1 = -12 , an = an-1 - 8
Special Sequences
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69 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 10 , an = an-1 + 8
Special Sequences
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70 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = 10 + (n-1)(8)
an = 10(8)n-1
an = 8 + (n-1)(10)
an = 8(10)n-1
a1 = 10 , an = an-1 + 8
Special Sequences
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71 Identify the sequence as arithmetic, geometric,or neither.
A Arithmetic
B Geometric
C Neither
a1 = 24 , an = (1/2)an-1
Special Sequences
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72 Which equation could be used to find the nth term of the recursive formula directly?
A
B
C
D
an = 24 + (n-1)(1/2)
an = 24(1/2)n-1
an = (1/2) + (n-1)24
an = (1/2)(24)n-1
a1 = 24 , an = (1/2)an-1
Special Sequences
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Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
4
7
7 + 4 = 11
11 + 7 = 18
18 + 11 =29
Find the first five terms of the sequence:a1 = 4, a2 = 7, and an = an-1 + an-2
Special Sequences
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Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
6
8
2(8) + 3(6) = 34
2(34) + 3(8) = 92
2(92) + 3(34) = 286
Find the first five terms of the sequence:a1 = 6, a2 = 8, and an = 2an-1 + 3an-2
Special Sequences
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Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
10
6
2(6) -10 = 2
2(2) - 6 = -2
2(-2) - 2= -6
Find the first five terms of the sequence:a1 = 10, a2 = 6, and an = 2an-1 - an-2
Special Sequences
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Special Recursive Sequences
Some recursive sequences not only rely on the preceding term, but on the two preceding terms.
1
1
1+1 = 2
1 + 2 = 3
2 + 3 =5
Find the first five terms of the sequence:a1 = 1, a2 = 1, and an = an-1 + an-2
Special Sequences
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The sequence in the preceding example is called
The Fibonacci Sequence1, 1, 2, 3, 5, 8, 13, 21, . . .
where the first 2 terms are 1'sand any term there after is the sum of preceding two terms.
This is as famous as a sequence can get and is worth remembering.
Special Sequences
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73 Find the first four terms of sequence:
A
B
C
D
a1 = 5, a2 = 7, and an = a1 + a2
7, 5, 12, 19
5, 7, 35, 165
5, 7, 12, 19
5, 7, 13, 20
Special Sequences
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74 Find the first four terms of sequence:
A
B
C
D
a1 = 4, a2 = 12, and an = 2an-1 - an-2
4, 12, -4, -20
4, 12, 4, 12
4, 12, 20, 28
4, 12, 20, 36
Special Sequences
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75 Find the first four terms of sequence:
A
B
C
D
a1 = 3, a2 = 3, and an = 3an-1 + an-2
3, 3, 6, 9
3, 3, 12, 39
3, 3, 12, 36
3, 3, 6, 21
Special Sequences
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Slide 130 / 145
Look for a pattern when powering a binomial.
Binomial Theorem
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One pattern comes from the coefficients.
11 1
1 2 11 3 3 1
1 4 6 4 11 5 10 10 5 1
1 6 15 20 15 6 11 7 21 35 35 21 7 1
This is Pascal's Triangle.
Binomial Theorem
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Example: Find the coefficient of the 4 term of 6th power of a binomial.
Binomial Theorem
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76 Calculate
Binomial Theorem
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77 Calculate
Binomial Theorem
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78 Calculate
Binomial Theorem
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79 What is the binomial coefficient of the 4th term of x + y to the 8th power?
Binomial Theorem
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80 What is the binomial coefficient of the 2nd term of x + y to the 3rd power?
Binomial Theorem
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The second pattern of powering a binomial comes from the exponents.
Click to see the Binomial Expansion.
But what if instead of x + y we have 2x -3?
Binomial Theorem
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The Binomial Theorem
Binomial Theorem
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Example: Expand (2x -3)4
Binomial Theorem
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81 What is the coefficient of the 4th term (2x - 3) to the 5th power?
Binomial Theorem
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82 What is the coefficient of the 3rd term (2x - 3) to the 6th power?
Binomial Theorem
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83 What is the coefficient of the 6th term (2x - 3) to the 7th power?
Binomial Theorem
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