Arithmetic Sequences and Series-Boger
Arithmetic Sequences and Series-Boger
Arithmetic Sequences and Series-Boger
An introduction…………
1, 4, 7, 10, 13 35 2, 4, 8, 16, 32 62
9, 1, 7, 15 12 9, 3, 1, 1/ 3 20 / 3
6.2, 6.6, 7, 7.4 27.2 1, 1/ 4, 1/16, 1/ 64 85 / 64
, 3, 6 3 9 , 2.5, 6.25 9.75
Arithmetic Sequences Geometric Sequences
ADD MULTIPLY
To get next term To get next term
n
DOMAIN: 1 2 3 4 5 The domain gives
the relative position
of each term.
SOLUTION
(1) 1
a1 1st term
(1) 1 2
( 2) 2
a2 2nd term
(2) 1 3
(3) 3
a3 3rd term
(3) 1 4
(4) 4
a4 4th term
(4) 1 5
(5) 5
a5 5th term
(5) 1 6
Writing Terms of Sequences
Arithmetic Sequence, d = 7
21, 28, 35, 42
an = a1 + (n – 1)d
an a1 (n 1)d
a14 4 (14 1)3
4 (13)3
4 39
43
Try this one: Find a16 if a1 1.5 and d 0.5
1.5 a1 First term
a16 an nth term
16 n number of terms
NA Sn sum of n terms
0.5 d common difference
an a1 n 1 d
a16 = 1.5+7.5
a16 = 9
The table shows typical costs for a construction company to rent a
crane for one, two, three, or four months. Assuming that the arithmetic
sequence continues, how much would it cost to rent the crane for
twelve months?
Months Cost ($) 75,000 a1 First term an a1 n 1 d
1 75,000 a12 an nth term a12 = 75,000 + (12 - 1)(15,000)
2 90,000
3 105,000 12 n number of terms a12 = 75,000 + (11)(15,000)
4 120,000 NA Sn sum of n terms
a12 = 75,000+165,000
15,000 d common difference
a12 = $240,000
In the arithmetic sequence
4,7,10,13,…, which term has a
value of 301?
an a1 (n 1)d
301 4 ( n 1)3
301 4 3n 3
301 1 3n
300 3n
100 n
Try this one: Find n if an 633, a1 9, and d 24
9 a1 First term
633 an nth term
n n number of terms
NA Sn sum of n terms
24 d common difference
an a1 n 1 d
633 = 9 + (n - 1)(24)
633 = 9 + 24n - 24
633 = 24n – 15
648 = 24n
n = 27
Given an arithmetic sequence with a15 38 and d 3, find a1.
a1 a1 First term
38 an nth term
15 n number of terms
NA Sn sum of n terms
-3 d common difference
an a1 n 1 d
38 = a1 + (15 - 1)(-3)
38 = a1 + (14)(-3)
38 = a1 - 42
a1 = 80
Find d if a1 6 and a 29 20
-6 a1 First term
20 an nth term
29 n number of terms
NA Sn sum of n terms
d d common difference
an a1 n 1 d
20 = -6 + (29 - 1)(d)
20 = -6 + (28)(d)
26 = 28d
13
d
14
Write an equation for the nth term of the arithmetic
sequence 8, 17, 26, 35, …
8 a1 First term
9 d common difference
an a1 n 1 d
an = 8 + (n - 1)(9)
an = 8 + 9n - 9
an = 9n - 1
Arithmetic Mean: The terms between any two
nonconsecutive terms of an arithmetic sequence.
-4 a1 First term an a1 n 1 d
5 an nth term 5 = -4 + (4 - 1)(d)
4 n number of terms 5 = -4 + (3)(d)
NA Sn sum of n terms 9 = (3)(d)
d d common difference d=3
21 a1 First term an a1 n 1 d
45 an nth term 45 = 21 + (5 - 1)(d)
5 n number of terms 45 = 21 + (4)(d)
NA Sn sum of n terms 24 = (4)(d)
d d common difference d=6
Example:
3, 5, 7, 9, 11, 13 3 + 5 + 7 + 9 + 11 + 13
Recall
Vocabulary of Sequences (Universal)
a1 First term
an nth term
n number of terms
Sn sum of n terms
d common difference
Find the sum of the first 63 terms of the arithmetic sequence -19, -13, -7,…
-19 a1 First term
353 ?? an nth term
63 n number of terms
Sn Sn sum of n terms
6 d common difference
n n
Sn a1 an an a1 n 1 d Sn a1 an
2 2
63
S63 19 353
2
a63 = 353
S63 10521
Find the first 3 terms for an arithmetic series in which a1 = 9, an = 105, Sn =741.
9 a1 First term
105 an nth term
13 ?? n number of terms
9, 17, 25
741 Sn sum of n terms
?? d common difference
n
an a1 n 1 d Sn a1 an an a1 n 1 d
2
A radio station considered giving away $4000 every day in the month of August for a
total of $124,000. Instead, they decided to increase the amount given away every day
while still giving away the same total amount. If they want to increase the amount by
$100 each day, how much should they give away the first day?
a1 First term n
a1 Sn a1 an
2
?? an nth term
31 days n number of terms
$124,000 Sn sum of n terms
$100/day d common difference
n
Sn a1 an an a1 n 1 d
2
Sigma Notation ( )
B
SIGMA
(SUM OF TERMS) a
n A
n NTH TERM
(SEQUENCE)
j 2 1 2 2 2 3 2 4 2 18
j1
2a 2 4 2 5 2 6 2 7 44
a 4
If the sequence is arithmetic (has a common difference) you can use the Sn formula
4
n
j 2
j1
1+2=3 a1 First term Sn a1 an
2
4+2=6 an nth term
4 n number of terms
?? Sn sum of n terms
NA d common difference
= 90
10 + 17 + 26 + 37
No, there is no common difference
Thus you cannot use the Sn formula.
= 2.71828
Rewrite using sigma notation: 3 + 6 + 9 + 12
Arithmetic, d= 3
an a1 n 1 d
an 3 n 1 3
an 3n
4
3n
n1