Download Aim: What is an arithmetic sequence and series?

Aim: What is an arithmetic sequence and
series?
Do Now: Find the next three numbers in
the sequence 1, 1, 2, 3, 5, 8, . . .
sequence – a set of ordered numbers
Fibonacci sequence, first published in book
titled, Liber abaci, in 1202 by Leonardo of Pisa.
Dealt with reproductive rights of rabbits.
Leonardo also introduced algebra in Europe
from the mideast. Algebra was occasionally
referred to as Ars Magna, “the Great Art”.
recursive – 1 or more of the first terms are given –
all other terms are defined by using the previous
terms
pattern found in nature
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Fibonacci Patterns
Suppose a newly-born pair of rabbits, one male, one
female, are put in a field. Rabbits are able to mate at the
age of one month so that at the end of its second month a
female can produce another pair of rabbits. Suppose that
our rabbits never die and that the female always produces
one new pair (one male, one female) every month from the
second month on. The puzzle that Fibonacci posed was...
Aim: Arithmetic
Sequence
Course: Alg. 2 & Trig.
How many pairs
will there
be in one year?
Fibonacci Patterns
If we take the ratio of two successive numbers in
Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide
each by the number before it, we will find the following
series of numbers:
1/1 = 1, 2/1 = 2, 3/2 = 1·5, 5/3 = 1·666...,
8/5 = 1·6, 13/8 = 1·625, 21/13 = 1·61538...
approached the
Golden Ratio - Course:
 Alg. 2 & Trig.
Aim: Arithmetic Sequence
Sequence
4, 8, 12, 16, 20,
. . . .24, . . .
If the pattern is extended,
what are the next two terms?
How is this sequence different from
the famous Fibonacci sequence?
4, 8, 12, 16, 20, 24, . . .
4 4 4
4
positive integers 1
2
terms of
sequence
8 12 16
4
Aim: Arithmetic Sequence
4
3
4
n
4n
Course: Alg. 2 & Trig.
Model Problem
Write the rule that can be used in forming a
sequence 1, 4, 9, 16, . . . , then use the rule to
find the next three terms of the sequence.
positive integers 1
2
3
4
n
terms of
sequence
4
9
16
n2
1
1, 4, 9, 16, 25, 36, 49
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem
Write the first five terms of the sequence
where rule for the nth term is represented by
n+2
1 2 3 4 5
n
positive integers 1 2 3 4 5
n
terms of
sequence
3
Aim: Arithmetic Sequence
4
5
6 7
n+2
Course: Alg. 2 & Trig.
Definition of Arithmetic Sequence
A sequence is arithmetic if the differences
between consecutive terms are the same.
Sequence
1 term
n term
a1, a2, a3, a4, . . . . . an, . . .
is arithmetic if there is a number d such that
a2 – a1 = d, a3 – a2 = d, a4 – a3 = d, etc.
The number d is the common difference on
the arithmetic sequence. Each term after the
first is the sum of the preceding term and a
constant, c.
st
th
7, 11, 15, 19, . . . . 4n + 3, . . .
4 4 4
4=d
finite
2, -3, -8, -13, . . . . 7 – 5n, . . .
-5 -5 -5
-5Alg.
= 2d& Trig.
Aim: Arithmetic Sequence
Course:
infinite
The nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence has
the form
an = dn + c
where d is the common difference between
consecutive terms of the sequence and
c = a1 – d
7, 11, 15, form
19, . . of
. . the
4n +nth
3, term
. . . is
An alternative
4 a4n =4a1 + (n4–=1)d
d
Find a formula for the nth term of the
arithmetic sequence whose common
difference is 3 and whose first term is 2.
a1 = 2 an = dn + c 2 = 3(1) + c c = -1
an = dn + c
an = 3n – 1
2, 5, 8, 11, 14, . . . , 3n – 1, . . .
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem
Find the twelfth term of the arithmetic
sequence 3, 8, 13, 18, . . . .
an = dn + c
d=5
c = a1 – d
a12 = ?
an = a1 + (n – 1)d
a12 = 3 + (12 – 1)5
a12 = 3 + (11)5 = 58
Rule?
an = dn + c
c = a1 – d
an = 5n – 2
c = 3 – 5 = -2
check: 18 = 5(4) – 2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem
The fourth term of an arithmetic sequence is
20, and the 13th term is 65. Write the first
several terms of this sequence.
an = dn + c
a13 = 9d + a4
65 = 9d + 20
5=d
an = dn + c
a4 = 5(4) + c
20 = 5(4) + c
0=c
an = 5n + 0
1
2
3
4
5
6
5, 10, 15, 20, 25, 30, . . .
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Regents Problem
Find the first four terms of the recursive
sequence defined below.
a1 = -3 an = a(n – 1) – n
a2 = a(2 – 1) – 2
a3 = a(3 – 1) – 3
a2 = a(1) – 2
a3 = a(2) – 3
a2 = -3 – 2 = -5
a3 = -5 – 3 = -8
a4 = a(4 – 1) – 4
a4 = a(3) – 4
a4 = -5 – 4 = -12
-3, -5, -8, -12
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Aim: What is an arithmetic sequence and
series?
Do Now:
Regents Problem
What is the 10th term of the
arithmetic sequence -1, 3, 7, 11, . . .
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Regents Problem
What is the 10th term of the arithmetic
sequence -1, 3, 7, 11, . . .
d=4
an = a1 + (n – 1)d
a10 = -1 + (10 – 1)4
a10 = -1 + (9)5 = 44
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Arithmetic Series
A series is the expression for the sum of the
terms of a sequence.
finite sequence
finite series
6, 9, 12, 15, 18
6 + 9 + 12 + 15 + 18
a1, a2, a3, a4, a5
a1 + a 2 + a3 + a4 + a5
infinite sequence
infinite series
3, 7, 11, 15, . . .
3 + 7 + 11 + 15 + . . .
a1, a2, a3, a4, . . .
a1 + a 2 + a3 + a4 + . . .
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
The Sum of an Arithmetic Sequence: Series
When
The
sum
famous
of a finite
German
arithmetic
mathematician
sequenceKarl
with
n termswas
Gauss
is a child, his
n teacher required the
S  (a1  an )
students to find the2sum of the first 100
natural
numbers.
The
teacher
expected
An arithmetic
series
is the
indicated
sumthis
of
problem
the class busy
for some
the termstoofkeep
an arithmetic
sequence.
time. Gauss gave the answer almost
Find the following sum
immediately. Can you?
1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
Is this an arithmetic sequence? Why? d = 2
n
10 terms
S  (a1  an )
2
n = 10
a1 = 1
an = 19
10
S  (1  19)  100
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
The Sum of an Arithmetic Sequence: Series
In an arithmetic series, if a1 is the first term,
n is the number of terms, an is the nth term,
and d is the common difference, then Sn the
sum of the arithmetic series, is given by the
formulas:
n
S  (a1  an )
2
or
n
S  [2a1  ( n  1)d ]
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem
Find the sum of the first ten terms of an
arithmetic sequence whose first term is 5
and whose 10th term is -13.
n
S  (a1  an )
2
a1 = 5
a10 = -13
10
S   5   13    5  8   40
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem
Find the sum of the first fifty terms of an
arithmetic sequence 3 + 5 + 7 + 9 + . . . .
n
S  (a1  an )
2
a1 = 5
a50 = ?
d = 2, n = 50
an = dn + c
c = a1 – d
an = a1 + (n – 1)d
an = a1 + (n – 1)d
an = 3 + (50 – 1)2 = 101
50
S   3  101  2600
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Model Problem – Option 2
Find the sum of the first fifty terms of an
arithmetic sequence 3 + 5 + 7 + 9 + . . . .
n
S  [2a1  ( n  1)d ]
2
a1 = 5
d = 2, n = 50
50
S  [2  3   (50  1)2]  2600
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Application
A small business sells $10,000 worth of
products during its first year. The owner of
the business has set a goal of increasing
annual sales by $7,500 each year for 9 years.
Assuming that this goal is met, find the total
sales during the first 10 years this business is
in operation.
a1 = 10000 d = 7500 c = 10000 – 7500 = 2500
n
S  (a1  an )
2
an = 7500n + 2500
a10 = 7500(10)+ 2500 = 77500
10
S  (10000  77500)  $437,500
2
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.
Application
A man wishes to pay off a debt of $1,160 by
making monthly payments in which each payment
after the first is $4 more than that of the previous
month. According to this plan, how long will it
take him to pay the debt if the first payment is $20
and no interest is charged?
Sn = 1160
d=4
a1 = $20
n=?
n
n
1160   2  20    n  1 4    40   4n  4  
2
2
n
  4n  36  2n2  18n  1160
2
2n2  18n  1160  0
n2  9n  580  0
n  29, n  20
 n  29  n  20   0
n
S  [2a1  ( n  1)d ]Course: Alg. 2 & Trig.
2
Aim: Arithmetic Sequence
Arithmetic Means
The terms between any two nonconsecutive
terms of an arithmetic sequence are called
arithmetic means. In the sequence below, 38
and 49 are the arithmetic means between 27
and 60.
5, 16, 27, 38, 49, 60
arithmetic
means
between
27 & 60
simple example: insert one arithmetic mean
between 16 and 20
an = a1 + (n – 1)d
20 = 16 + (2)d
an = a3 = 20
16, 18, 20
d=2
Aim: Arithmetic Sequence
a1 = 16
(n – 1) = 3 – 1 = 2
Course: Alg. 2 & Trig.
Model Problem
Write an arithmetic sequence that has five
arithmetic means between 4.9 and 2.5.
4.5 ___,
4.1 3.7
2.9 2.5
4.9, ___,
___, 3.3
___, ___,
an = a1 + (n – 1)d
n=7
an = a7 = 2.5
a1 = 4.9
2.5 = 4.9 + (6)d
d = -0.4
a2 = 4.9 + (-0.4) = 4.5
a3 = 4.5 + (-0.4) = 4.1
a4 = 4.1 + (-0.4) = 3.7
Aim: Arithmetic Sequence
a5 = 3.7 + (-0.4) = 3.3
a6 = 3.3 + (-0.4) = 2.9
Course: Alg. 2 & Trig.
Model Problem
Mrs. Gonzales sells houses and makes a commission
of $3750 for the first house sold. She will receive a
$500 increase in commission for each additional
house sold. How many houses must she sell to reach
total commissions of $65000?
Sn = 65000 a1 = 3750 d = 500
n
S  (a1  an )
2
an = a1 + (n – 1)d
n
65000  ( 3750  an )
2
n
65000  ( 3750  (3750  ( n  1)500)
2
2
130000  7000n  500n
n = 10.58 and –24.58
She must sell 11 houses.
Aim: Arithmetic Sequence
Course: Alg. 2 & Trig.