# Arithmetic Sequence and Series

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This ppt. provides an introduction to Arithmetic Sequence and Series
1. Arithmetic Sequences and Series
2. 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
To get next term To get next term
Arithmetic Series Geometric Series
Sum of Terms Sum of Terms
3. USING AND WRITING SEQUENCES
The numbers in sequences are called terms.
You can think of a sequence as a function whose domain
is a set of consecutive integers. If a domain is not
specified, it is understood that the domain starts with 1.
4. USING AND WRITING SEQUENCES
n
DOMAIN: 1 2 3 4 5 The domain gives
the relative position
of each term.
The range gives the
an
RANGE: 3 6 9 12 15
terms of the sequence.
This is a finite sequence having the rule
an = 3n,
where an represents the nth term of the sequence.
5. Writing Terms of Sequences
Write the first five terms of the sequence an = 2n + 3.
a 1 = 2(1) + 3 = 5 1st term
a 2 = 2(2) + 3 = 7 2nd term
a 3 = 2(3) + 3 = 9 3rd term
a 4 = 2(4) + 3 = 11 4th term
a 5 = 2(5) + 3 = 13 5th term
6. Writing Terms of Sequences
n
Write the first five terms of the sequence an  n 1
.
(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
7. Writing Terms of Sequences
Write the first five terms of the sequence an = n! - 2.
a 1 = (1)!-2 = -1 1st term
a 2 = (2)! - 2 = 0 2nd term
a 3 = (3)! - 2 = 4 3rd term
a 4 = (4)! - 2 = 22 4th term
a 5 = (5)! - 2 = 118 5th term
8. Writing Terms of Sequences
Write the first five terms of the sequence f (n) = (–2) n – 1 .
f (1) = (–2) 1 – 1 = 1 1st term
f (2) = (–2) 2 – 1 = –2 2nd term
f (3) = (–2) 3 – 1 = 4 3rd term
f (4) = (–2) 4 – 1 = – 8 4th term
f (5) = (–2) 5 – 1 = 16 5th term
9. Arithmetic Sequences and Series
Arithmetic Sequence: sequence whose
consecutive terms have a common
difference.
Example: 3, 5, 7, 9, 11, 13, ...
The terms have a common difference of 2.
The common difference is the number d.
To find the common difference you use an+1
– an
Example: Is the sequence arithmetic?
–45, –30, –15, 0, 15, 30
Yes, the common difference is 15
10. Find the next four terms of –9, -2, 5, …
Arithmetic Sequence
-2 - -9 = 7 and 5 - -2 = 7
7 is referred to as the common difference (d)
Common Difference (d) – what we ADD to get next term
Next four terms……12, 19, 26, 33
11. Find the next four terms of 0, 7, 14, …
Arithmetic Sequence, d = 7
21, 28, 35, 42
Find the next four terms of -3x, -2x, -x, …
Arithmetic Sequence, d = x
0, x, 2x, 3x
Find the next four terms of 5k, -k, -7k, …
Arithmetic Sequence, d = -6k
-13k, -19k, -25k, -31k
12. How do you find any term in this sequence?
To find any term in an arithmetic sequence, use the formula
an = a1 + (n – 1)d
where d is the common difference.
13. Vocabulary of Sequences (Universal)
a1  First term
an  nth term
n  number of terms
Sn  sum of n terms
d  common difference
nth term of arithmetic sequence  an a1   n  1 d
n
sum of n terms of arithmetic sequence  Sn   a1  an 
2
14. Find the 14th term of the
arithmetic sequence
4, 7, 10, 13,……
an a1  (n  1)d
a14  4 (14  1)3
4  (13)3
4  39
43
15. 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 + (16 - 1)(0.5)
a16 = 1.5 + (15)(0.5)
a16 = 1.5+7.5
a16 = 9
16. 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
17. 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
18. 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
19. 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
20. 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
21. 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
22. Arithmetic Mean: The terms between any two
nonconsecutive terms of an arithmetic sequence.
Ex. 19, 30, 41, 52, 63, 74, 85, 96
41, 52, 63 are the Arithmetic Mean between 30 and 74
23. Find two arithmetic means between –4 and 5
-4, ____, ____, 5
-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
The two arithmetic means are –1 and 2, since –4, -1, 2, 5
forms an arithmetic sequence
24. Find three arithmetic means between 21 and 45
21, ____, ____, ____, 45
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
The three arithmetic means are 27, 33, and 39
since 21, 27, 33, 39, 45 forms an arithmetic sequence
25. Arithmetic Series: An indicated sum of terms in an arithmetic
Arithmetic Sequence VS Arithmetic Series
3, 5, 7, 9, 11, 13 3 + 5 + 7 + 9 + 11 + 13
26. Recall
Vocabulary of Sequences (Universal)
a1  First term
an  nth term
n  number of terms
Sn  sum of n terms
d  common difference
27. 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
28. 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
29. 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
30. Sigma Notation (  )
Used to express a series or its sum in abbreviated form.
31. UPPER LIMIT
(NUMBER)
B
SIGMA
(SUM OF TERMS) a
nA
n NTH TERM
(SEQUENCE)
INDEX LOWER LIMIT
(NUMBER)
32. 4
  j  2  1  2   2  2    3  2   4  2
j1
18
7
  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
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
33. = 90
Is the sequence arithmetic?
10 + 17 + 26 + 37
No, there is no common difference
Thus you cannot use the Sn formula.
= 2.71828
34. 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
n1