Transcript 11.1-11.2: Arithmetic Sequences & Series
11.2: Arithmetic Sequences & Series
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n = a 1 + (n – 1)d
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n = a 1 + (
n
– 1)d
a
10 = a 1 + ( 10 – 1)d
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n =
a
1 + (n – 1)d
a
10 = 55 + (10 – 1)d
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n
a
10 = a 1 + (n – 1)
d
= 55 + (10 – 1) (-6)
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n
a
10 = a 1 + (n – 1)d = 55 + (10 – 1)(-6)
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37
a
1
a
2
a
3
a
4
a) Find the 10 th term in the sequence.
a
n
a
10 = a 1 + (n – 1)d = 55 + ( 10 – 1 )(-6)
a
10 = 55 + ( 9 )(-6)
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37 a) Find the 10 th
a
1
a
2
a
3
a
4
term in the sequence.
a
n
a
10 = a 1 + (n – 1)d = 55 + (10 – 1)(-6)
a
10
a
10 = 55 + = 55 (9)(-6) – 54
n
th Term of an Arithmetic Sequence:
a n
= a 1 + (n – 1)d
Ex. 1Determine the following using the table below.
55 49 43 37 a) Find the 10 th
a
1
a
2
a
3
a
4
term in the sequence.
a
n
a
10 = a 1 + (n – 1)d = 55 + (10 – 1)(-6)
a
10 = 55 + (9)(-6)
a
10 = 55 – 54
a
10 = 1
b) Write an equation for the
n
th term of the sequence.
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
a
n = 55 + (n – 1) (-6)
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
a
n = 55 + (n – 1) (-6)
a
n = 55 - 6 (n – 1)
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
a
n = 55 + (n – 1)(-6)
a
n = 55 - 6(n – 1)
a
n = 55 - 6n + 6
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
a
n = 55 + (n – 1)(-6)
a
n = 55 - 6(n – 1)
a
n = 55 - 6n + 6
a
n = - 6n + 61
b) Write an equation for the
n
th term of the sequence.
a
n = a 1 + (n – 1)d
a
n = 55 + (n – 1)(-6)
a
n = 55 - 6(n – 1)
a
n = 55 - 6n + 6
a
n = - 6n + 61
Ex. 2 Find the arithmetic means in the sequence below.
24, ___, ___, ___, ___, -1
Ex. 2 Find the arithmetic means in the sequence below.
24, ___, ___, ___, ___, -1 ***Find the missing terms in the sequence!
Ex. 2 Find the arithmetic means in the sequence below.
24, ___, ___, ___, ___, -1
a
1
a
2
a
3
a
4
a
5
a
6
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
a
1
a
2
a
3
a
4
a
5
a
6
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
a
1
a
2
a
3
a
4
a
5
a
6
n
= 6
a
1 = 24
a
6 = -1
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
a
1
a
2
a
3
a
4
a
5
a
6
n
= 6
a
1 = 24
a
6 = -1
a
n = a 1 + (n – 1)d
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
a
1
a
2
a
3
a
4
a
5
a
6
n
= 6
a
1 = 24
a
6 = -1
a
n = a 1 + (n – 1)d -1 = 24 + ( 6 – 1)d
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
-1 =
a
3
24
a
4
a
1 = 24
a
n = a 1
a
6 = -1
+ (n – 1)d + ( 6 – 1)d
-1 = 24 + 5
d
-25 = 5
d
-5 =
d a
5
a
6
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
4
a
5
a
1 = 24
a
n = a 1
a
6 = -1
+ (n – 1)d -1 = 24 + ( 6 – 1)d
a
6 -5 =
d a
1 = 24
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
4
a
5
a
1 = 24
a
n = a 1
a
6 = -1
+ (n – 1)d -1 = 24 + ( 6 – 1)d
a
6 -5 =
d a
1 = 24
a
2 = 24 + (-5) = 19
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
4
a
5
a
1 = 24
a
n = a 1
a
6 = -1
+ (n – 1)d -1 = 24 + ( 6 – 1)d
a
6 -5 =
d a
1 = 24
a
2 = 24 + (-5) = 19
a
3 = 19 + (-5) = 14
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
1 = 24
a
4
a
5
a
6 = -1
a
1 = 24
a
2 = 24 + (-5) = 19
a
3 = 19 + (-5) = 14
a
4 = 14 + (-5) = 9
a
6
d
= -5
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
1 = 24
a
4
a
5
a
6 = -1
a
1 = 24
a
2 = 24 + (-5) = 19
a
3 = 19 + (-5) = 14
a
4 = 14 + (-5) = 9
a
5 = 9 + (-5) = 4
a
6
d
= -5
Ex. 2 Find the arithmetic means in the sequence below.
24 , ___, ___, ___, ___, -1
n
= 6
a
1
a
2
a
3
a
1 = 24
a
4
a
5
a
6 = -1
a
1 = 24
a
2 = 24 + (-5) = 19
a
3 = 19 + (-5) = 14
a
4 = 14 + (-5) = 9
a
5 = 9 + (-5) = 4
a
6
d
= -5
Sum of an Arithmetic Series
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
S
n = ½n[ 2a 1 + (n – 1)d ]
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2
a
1 = ½n[ a 1 + (n +
a
n ] – 1)d ]
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2
a
1 = ½n[
a
1 + (n + a n ] – 1)d ]
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following:
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following: a)
a
1
= 58,
a
n
= -7,
n
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½
n
[
a
1 + (n +
a
n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following: a)
a
1
= 58,
a
n
= -7,
n
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following: a)
a
1
S
n
= 58,
a
n = ½n[ a 1
= -7,
+ a n ]
n
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following: a)
a
1
S
n
= 58 ,
a
n = ½n[ a 1
= -7
+ a n ]
,
n S
n = ½( 26 )[ 58 - 7 ]
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S
n
S
n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
for each of the following: a)
a
1
S
n
= 58,
a
n = ½n[ a 1
= -7,
+ a n ]
n S
n = ½(26)[ 58 - 7 ]
S
n = ½(26)[ 51 ]
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S S
n n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
a)
a
1
S
n
= 58,
a
n = ½n[ a 1
= -7,
+ a n ]
n S
n = ½(26)[ 58 - 7 ]
S
n = ½(26) [ 51 ]
S
n
for each of the following:
= 13 [ 51 ]
= 26
Sum of an Arithmetic Series
The sum
S
n
of the first
n
terms of an arithmetic series is given by the following:
OR
S S
n n = ½n[ 2a 1 = ½n[ a 1 + (n + a n ] – 1)d ]
Ex. 3 Find
S
n
a)
a
1
S
n
= 58,
a
n = ½n[ a 1
= -7,
+ a n ]
n S
n = ½(26)[ 58 - 7 ]
S
n = ½(26)[ 51 ]
S
n
for each of the following:
= 13(51) = 663
= 26
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
= 16
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
a
1
= 16 = 4(1) – 2 = 2
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
a
1
a
n
= 16 = 4(1) – 2 = 2 = 4(16) – 2 = 62
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
a
1
a
n
= 16 = 4(1) – 2 = 2 = 4(16) – 2 = 62
S
n = ½n[ a 1 + a n ]
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
a
1
a
n
= 16 = 4(1) – 2 = 2 = 4(16) – 2 = 62
S
n
S
n
= ½n[ a 1
= ½( 16 )[ 2
+ a n
+ 62
]
]
Ex. 4 16 ∑ (4
k
– 2)
k
= 1
n
a
1
a
n
= 16 = 4(1) – 2 = 2 = 4(16) – 2 = 62
S
n
S
n
S
n
= ½n[ a 1
= ½( 16 )[ 2
+ a n
+ 62
]
] = 512