1. 11.2 Arithmetic Sequences
Ephesians 1:7 "In him we have redemption through his blood,
the forgiveness of our trespasses, according to the riches of his
grace."
2. A sequence a1 , a2 , a3 , K is an
Arithmetic Sequence if there is a constant, d
for which an = an−1 + d for all integers n > 1 .
3. A sequence a1 , a2 , a3 , K is an
Arithmetic Sequence if there is a constant, d
for which an = an−1 + d for all integers n > 1 .
d is called the common difference
4. A sequence a1 , a2 , a3 , K is an
Arithmetic Sequence if there is a constant, d
for which an = an−1 + d for all integers n > 1 .
d is called the common difference
d = an − an−1
5. A sequence a1 , a2 , a3 , K is an
Arithmetic Sequence if there is a constant, d
for which an = an−1 + d for all integers n > 1 .
d is called the common difference
d = an − an−1
examples:
8, 16, 24, 32, K a1 = 8 an = an−1 + 8
6. A sequence a1 , a2 , a3 , K is an
Arithmetic Sequence if there is a constant, d
for which an = an−1 + d for all integers n > 1 .
d is called the common difference
d = an − an−1
examples:
8, 16, 24, 32, K a1 = 8 an = an−1 + 8
10, 7, 4, 1, − 2, − 5, K a1 = 10 an = an−1 + ( −3)
9. Determine which sequences are Arithmetic and
identify the Common Difference.
1) 5, 10, 15, 20, 25, K
yes d = 5
10. Determine which sequences are Arithmetic and
identify the Common Difference.
1) 5, 10, 15, 20, 25, K
yes d = 5
2) 1, − 2, 4, − 8, 16, − 32, K
11. Determine which sequences are Arithmetic and
identify the Common Difference.
1) 5, 10, 15, 20, 25, K
yes d = 5
2) 1, − 2, 4, − 8, 16, − 32, K
no
12. Determine which sequences are Arithmetic and
identify the Common Difference.
1) 5, 10, 15, 20, 25, K
yes d = 5
2) 1, − 2, 4, − 8, 16, − 32, K
no
3) 16, 9, 2, − 5, − 12, K
13. Determine which sequences are Arithmetic and
identify the Common Difference.
1) 5, 10, 15, 20, 25, K
yes d = 5
2) 1, − 2, 4, − 8, 16, − 32, K
no
3) 16, 9, 2, − 5, − 12, K
yes d = −7
14. Determine which sequences are Arithmetic and
identify the Common Difference.
4) 1, 1.5, 2, 2.5, 3, K
15. Determine which sequences are Arithmetic and
identify the Common Difference.
4) 1, 1.5, 2, 2.5, 3, K
yes d = .5
16. Determine which sequences are Arithmetic and
identify the Common Difference.
4) 1, 1.5, 2, 2.5, 3, K
yes d = .5
5) 1, 10, 11, 21, 32, 53, K
17. Determine which sequences are Arithmetic and
identify the Common Difference.
4) 1, 1.5, 2, 2.5, 3, K
yes d = .5
5) 1, 10, 11, 21, 32, 53, K
no
21. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
22. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
20 = a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
23. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
20 = a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
25 = a5 = a1 + 4d
24. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
20 = a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
25 = a5 = a1 + 4d
an =
25. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
20 = a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
25 = a5 = a1 + 4d
an = a1 + ( n − 1) d
26. Consider this arithmetic sequence:
5, 10, 15, 20, 25, K
5 = a1 = a1
10 = a2 = a1 + d
15 = a3 = a2 + d = ( a1 + d ) + d = a1 + 2d
20 = a4 = a3 + d = ( a1 + 2d ) + d = a1 + 3d
25 = a5 = a1 + 4d
an = a1 + ( n − 1) d
27. Find the 53rd term of this arithmetic sequence:
3, 11, 19, 27, K
28. Find the 53rd term of this arithmetic sequence:
3, 11, 19, 27, K
d=8
29. Find the 53rd term of this arithmetic sequence:
3, 11, 19, 27, K
d=8
∴ a53 = 3 + ( 52 ) ( 8 )
= 419
30. Find the 19th term of this arithmetic sequence:
65, 54, 43, 32, K
31. Find the 19th term of this arithmetic sequence:
65, 54, 43, 32, K
d = −11
32. Find the 19th term of this arithmetic sequence:
65, 54, 43, 32, K
d = −11
∴ a19 = 65 + (18 ) ( −11)
= −133
33. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
34. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
example: Find the explicit formula for the
general term of the arithmetic sequence if
a3 = 15 and a8 = 50
35. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
example: Find the explicit formula for the
general term of the arithmetic sequence if
a3 = 15 and a8 = 50
50 − 15
d= =7
8−3
36. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
example: Find the explicit formula for the
general term of the arithmetic sequence if
a3 = 15 and a8 = 50
50 − 15
d= =7
8−3
∴ a1 = 15 − 7 − 7 = 1
37. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
example: Find the explicit formula for the
general term of the arithmetic sequence if
a3 = 15 and a8 = 50
50 − 15
d= =7 an = 1+ ( n − 1) 7
8−3
∴ a1 = 15 − 7 − 7 = 1
38. If any two terms of an arithmetic sequence are
known, the sequence is completely determined.
example: Find the explicit formula for the
general term of the arithmetic sequence if
a3 = 15 and a8 = 50
50 − 15
d= =7 an = 1+ ( n − 1) 7
8−3
an = 1+ 7n − 7
∴ a1 = 15 − 7 − 7 = 1 an = 7n − 6
39. Groups: Find the explicit formula for the
general term of the arithmetic sequence if
a4 = 59 and a9 = −11
40. Groups: Find the explicit formula for the
general term of the arithmetic sequence if
a4 = 59 and a9 = −11
59 − ( −11) 70
d= = = −14
4−9 −5
∴ a1 = 59 + 3(14 ) = 101
an = 101+ ( n − 1) ( −14 )
an = 101− 14n + 14
an = 115 − 14n
41. HW #5
“Don’t be afraid to give up the good to go for the great.”
Kenny Rogers
