91 sequences
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Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Sequences
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence may be listed as f1, f2 , f3 , …
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list,
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list, fn = the n’th number on the list,
A sequence is an ordered list of infinitely many numbers that may or may not have a pattern.
Definition: A sequence is the list of outputs f(n) of a function f where n = 1, 2, 3, 4, ….. We write f(n) as fn.
Sequences
Example A:1, 3, 5, 7, 9,… is the sequence of odd numbers.1, 4, 9, 16, 25,… is the sequence of square numbers.5, -2, , e2, -110, …is a sequence without an obvious pattern.
A sequence may be listed as f1, f2 , f3 , … f100 = 100th number on the list, fn = the n’th number on the list,fn-1 = the (n – 1)’th number on the list or the number before fn.
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25,
Sequences
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
Sequences
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10,
Sequences
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
Sequences
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)
Example B: a. For the sequence of square numbers 1, 4, 9, 16, … f3 = 9, f4= 16, f5 = 25, and a formula for fn is fn = n2.
b. For the sequence of even numbers 2, 4, 6, 8, … f3= 6, f4 = 8, f5 = 10, and a formula for fn is fn = 2*n
Sequences
c. For the sequence of odd numbers 1, 3, 5, … a general formula is fn= 2n – 1.
d. For the sequence of odd numbers with alternating signs -1, 3, -5, 7, -9, …fn = (-1)n(2n – 1)A sequence whose signs alternate is called an alternating sequence as in part d.
Summation Notation
In mathematics, the Greek letter “” (sigma) means “to add”.
Summation Notation
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,
Summation Notation
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s.
Summation Notation
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100,
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fkk = 1
100
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk k = 1
100
A variable which is called the “index” variable, in this case k.
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk k = 1
100
A variable which is called the “index” variable, in this case k.k begins with the bottom numberand counts up (runs) to the top number.
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk k = 1
100
A variable which is called the “index” variable, in this case k.k begins with the bottom numberand counts up (runs) to the top number.
The beginning number
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk k = 1
100
A variable which is called the “index” variable, in this case k.k begins with the bottom numberand counts up (runs) to the top number.
The beginning number
The ending number
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk = f1 f2 f3 … f99 f100k = 1
100
A variable which is called the “index” variable, in this case k.k begins with the bottom numberand counts up (runs) to the top number.
The beginning number
The ending number
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
Summation Notation
Given a list, let's say, of 100 numbers, f1, f2, f3,.., f100, their sum f1 + f2 + f3 ... + f99 + f100 may be written in the - notation as:
fk = f1+ f2+ f3+ … + f99+ f100k = 1
100
A variable which is called the “index” variable, in this case k.k begins with the bottom numberand counts up (runs) to the top number.
The beginning number
The ending number
In mathematics, the Greek letter “” (sigma) means “to add”. Hence, “x” means to add the x’s,“(x*y)” means to add the x*y’s. (Of course the x's and xy's have to be given in the context.)
fk =k=4
8
ai = i=2
5
xjyj = j=6
9
aj = j=n
n+3
Summation Notation
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = i=2
5
xjyj = j=6
9
aj = j=n
n+3
Summation Notation
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = j=6
9
aj = j=n
n+3
Summation Notation
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = j=n
n+3
Summation Notation
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in mathematics.
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in mathematics. An example is the formula for averaging.
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in mathematics. An example is the formula for averaging. Given n numbers, f1, f2, f3,.., fn, their average (mean), written as f, is (f1 + f2 + f3 ... + fn-1 + fn)/n.
Example C:
fk = f4+ f5+ f6+ f7+ f8k=4
8
ai = a2+ a3+ a4+ a5i=2
5
xjyj = x6y6+ x7y7+ x8y8+ x9y9j=6
9
aj = an+ an+1+ an+2+ an+3j=n
n+3
Summation Notation
Summation notation are used to express formulas in mathematics. An example is the formula for averaging. Given n numbers, f1, f2, f3,.., fn, their average (mean), written as f, is (f1 + f2 + f3 ... + fn-1 + fn)/n. In notation, f =
k=1
n
fk
n
The index variable is also used as the variable that generates the numbers to be summed.
Summation Notation
Example D:
a. (k2 – 1) k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
Summation Notation
Example D:
a. (k2 – 1) = k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) k=3
5
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2)k=3
5
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17
k=3
5
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14
k=3
5
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14
k=3
5
In part b, the multiple (-1)k change the sums to an alternating sum, t
Example D:
a. (k2 – 1) = (52–1) + (62–1) + (72–1) + (82–1)
= 24 + 35 + 48 + 63 = 170
k=5
8
The index variable is also used as the variable that generates the numbers to be summed.
k=5 k=6 k=7 k=8
Summation Notation
b. (-1)k(3k + 2) =(-1)3(3*3+2)+(-1)4(3*4+2)+(-1)5(3*5+2) = -11 + 14 – 17 = -14
k=3
5
In part b, the multiple (-1)k change the sums to an alternating sum, that is, a sum where the terms alternate between positive and negative numbers.
Example D: