Transcript 12. 1 Arithmetic Sequences and Series

*
By the end of the section students will be able to find
determine whether a sequence is arithmetic, find the nth
term of an arithmetic sequence, find the partial sum of an
arithmetic series, as evidenced by completion of “I
have…who has…”.
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the nth term of
an arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
* What is the next number in each sequence?
1, 2, 3, 4, 5, …
2, 4, 6, …
6, 11, 16, 21, …
6, −2, −10, …
* These are all ARITHMETIC sequences because each term
ADD or SUBTRACTS the same amount.
* This amount is called the “common difference”
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
*Sequence = LIST
*Series = SUM
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
* 𝑎𝑛 means the nth term
* 𝑑 is the common difference
* 𝑆𝑛 is the sum of the first n terms.
Nth Term:
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
Partial Sum
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the nth term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
A.
47th term of −4, −1, 2, 5, …
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑎47 = 𝑎1 + 47 − 1 3
𝑎47 = −4 + 46 3
𝑎47 = −4 + 138
𝑎47 = 134
B.
1st term of a sequence whose 19th term is 42 and 𝑑 = −
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑎19 = 𝑎1 + 19 − 1 𝑑
2
42 = 𝑎1 + 18 −
3
42 = 𝑎1 − 12
𝑎1 = 54
2
3
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the nth term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
C. 41st term of a sequence in which 𝑎1 = 11, 𝑑 = −7
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑎41 = 𝑎1 + 41 − 1 𝑑
𝑎41 = 11 + 40 −7
𝑎41 = 11 − 280
𝑎41 = −269
D. 60th term of 9, 14, 19, …
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑎60 = 𝑎1 + 60 − 1 𝑑
𝑎60 = 9 + 59 5
𝑎60 = 9 + 295
𝑎60 = 304
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the nth term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
E.
Nth term of a sequence in which 𝑎1 = 3 and 𝑑 = −2
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎𝑛 = 3 + 𝑛 − 1 −2
𝑎𝑛 = 3 − 2𝑛 + 3
𝑎𝑛 = −2𝑛 + 6
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
A.
First 47 terms of the sequence −4, −1, 2, 5, …
𝑆𝑛 =
𝑆47
𝑛
𝑎1 + 𝑎𝑛
2
47
=
𝑎1 + 𝑎47
2
ℎ𝑜𝑤 𝑑𝑜 𝑤𝑒
𝑓𝑖𝑛𝑑 𝑡ℎ𝑖𝑠?
47
𝑆47 =
(−4 + 134)
2
47
𝑆47 =
130
2
𝑆47 = 47 65
𝑆47 = 3055
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
B.
First 19 terms of a sequence in which 𝑎1 = 54 and 𝑎19 =
42
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
19
𝑆19 =
54 + 42
2
19
𝑆19 =
96
2
𝑆19 = 19 48
𝑆19 = 912
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
C. First 60 terms of 9, 14, 19, …
𝑛
𝑎1 + 𝑎𝑛
2
60
=
𝑎1 + 𝑎60
2
𝑆𝑛 =
𝑆60
??
𝑆60 = 30 9 + 304
𝑆60 = 30 313
𝑆60 = 9390
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
A.
B.
C.
D.
E.
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
What is the 35th term in the sequence: 3, 6, 9, 12, …?
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
Determine the sum of the first 25 terms of the arithmetic
series 3 + 6 + 9 + 12 + ⋯
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
Find the sum of the first 20 terms of the series 2 + 4 + 6 +
8+⋯
𝑛
𝑆𝑛 = 𝑎1 + 𝑎𝑛
2
What is the 50th term in the sequence 2, 4, 6, 8, …?
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
An arithmetic sequence has 𝑎3 = 12 and 𝑎16 = 24.7, find 𝑎1 .
𝑎𝑛 = 𝑎𝑚 + 𝑛 − 𝑚 𝑑
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of “I have…who has…”.
1.
What is the 25th term in the sequence 8, 11, 14, 17, … ?
2.
What is the sum of the first 25 terms in the sequence 8,
11, 14, 17, … ?
3.
What is the 50th term for the sequence which has 𝑎1 =
− 2, 𝑑 = 7?
4.
What is 𝑑 if a sequence has a3 = 14, 𝑎14 = 58?
By the end of the section students will be able to find determine whether a sequence is arithmetic, find the n th term of an
arithmetic sequence, find the partial sum of an arithmetic series, as evidenced by completion of a graphic organizer.
* What is the 25th term in the sequence 8, 11, 14, 17, … ?
𝑎25 = 8 + 25 − 1 3 = 8 + 24 3 = 80
* What is the sum of the first 25 terms in the sequence 8, 11, 14, 17, … ?
𝑆25 =
25
25 ∙ 88
8 + 80 =
= 25 ∙ 44 = 1100
2
2
* What is the 50th term for the sequence which has 𝑎1 = −2, 𝑑 = 7?
𝑎50 = −2 + 50 − 1 7 = −2 + 49 7 = 341
* What is 𝑑 if a sequence has a3 = 14, 𝑎14 = 58?
𝑎14 = 𝑎3 + 14 − 3 𝑑
58 = 14 + 11 𝑑
44 = 11𝑑
𝑑=4