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: