11.2: Arithmetic Sequences

1
11.2 Arithmetic Sequences Series
2

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.

55 49 43 37
a1 a2 a3 a4
3

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.

55 49 43 37
a1 a2 a3 a4
4

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d

55 49 43 37
a1 a2 a3 a4
5

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 a1 (10 1)d

55 49 43 37
a1 a2 a3 a4
6

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)d

55 49 43 37
a1 a2 a3 a4
7

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)(-6)

55 49 43 37
a1 a2 a3 a4
8

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)(-6)

55 49 43 37
a1 a2 a3 a4
9

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)(-6)
• a10 55 (9)(-6)

55 49 43 37
a1 a2 a3 a4
10

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)(-6)
• a10 55 (9)(-6)
• a10 55 54

55 49 43 37
a1 a2 a3 a4
11

• nth Term of an Arithmetic Sequence
• an a1 (n 1)d
• Ex. 1Determine the following using the table
  below.
• a) Find the 10th term in the sequence.
• an a1 (n 1)d
• a10 55 (10 1)(-6)
• a10 55 (9)(-6)
• a10 55 54
• a10 1

55 49 43 37
a1 a2 a3 a4
12

• b) Write an equation for the nth term of the
  sequence.

13

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d

14

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d
• an 55 (n 1)(-6)

15

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d
• an 55 (n 1)(-6)
• an 55 - 6(n 1)

16

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d
• an 55 (n 1)(-6)
• an 55 - 6(n 1)
• an 55 - 6n 6

17

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d
• an 55 (n 1)(-6)
• an 55 - 6(n 1)
• an 55 - 6n 6
• an - 6n 61

18

• b) Write an equation for the nth term of the
  sequence.
• an a1 (n 1)d
• an 55 (n 1)(-6)
• an 55 - 6(n 1)
• an 55 - 6n 6
• an - 6n 61

19

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1

20

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• Find the missing terms in the sequence!

21

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6

22

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6

23

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6
• a1 24
• a6 -1

24

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6
• a1 24
• a6 -1
• an a1 (n 1)d

25

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6
• a1 24
• a6 -1
• an a1 (n 1)d
• -1 24 (6 1)d

26

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1
• an a1 (n 1)d
• -1 24 (6 1)d
• -1 24 5d
• -25 5d
• -5 d

27

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1
• an a1 (n 1)d
• -1 24 (6 1)d
• -5 d
• a1 24

28

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1
• an a1 (n 1)d
• -1 24 (6 1)d
• -5 d
• a1 24
• a2 24 (-5) 19

29

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1
• an a1 (n 1)d
• -1 24 (6 1)d
• -5 d
• a1 24
• a2 24 (-5) 19
• a3 19 (-5) 14

30

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1 d -5
• a1 24
• a2 24 (-5) 19
• a3 19 (-5) 14
• a4 14 (-5) 9

31

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1 d -5
• a1 24
• a2 24 (-5) 19
• a3 19 (-5) 14
• a4 14 (-5) 9
• a5 9 (-5) 4

32

• Ex. 2 Find the arithmetic means in the sequence
  below.
• 24, ___, ___, ___, ___, -1
• a1 a2 a3 a4 a5 a6
• n 6 a1 24 a6 -1 d -5
• a1 24
• a2 24 (-5) 19
• a3 19 (-5) 14
• a4 14 (-5) 9
• a5 9 (-5) 4

33

• Sum of an Arithmetic Series

34

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following

35

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d

36

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an

37

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an

38

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an

39

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following

40

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26

41

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26

42

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26
• Sn ½n a1 an

43

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26
• Sn ½n a1 an
• Sn ½(26) 58 - 7

44

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26
• Sn ½n a1 an
• Sn ½(26) 58 - 7
• Sn ½(26) 51

45

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26
• Sn ½n a1 an
• Sn ½(26) 58 - 7
• Sn ½(26) 51
• Sn 13 51

46

• Sum of an Arithmetic Series
• The sum Sn of the first n terms of an arithmetic
  series is given by the following
• Sn ½n 2a1 (n 1)d
• OR Sn ½n a1 an
• Ex. 3 Find Sn for each of the following
• a) a1 58, an -7, n 26
• Sn ½n a1 an
• Sn ½(26) 58 - 7
• Sn ½(26) 51
• Sn 13(51) 663

47

• Ex. 4 16
• ? (4k 2)
• k 1

48

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16

49

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16
• a1 4(1) 2 2

50

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16
• a1 4(1) 2 2
• an 4(16) 2 62

51

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16
• a1 4(1) 2 2
• an 4(16) 2 62
• Sn ½n a1 an

52

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16
• a1 4(1) 2 2
• an 4(16) 2 62
• Sn ½n a1 an
• Sn ½(16)2 62

53

• Ex. 4 16
• ? (4k 2)
• k 1
• n 16
• a1 4(1) 2 2
• an 4(16) 2 62
• Sn ½n a1 an
• Sn ½(16)2 62
• Sn 512