Download Slide 1 - Mrs. Hille`s FunZone

Exercise
1+2+3+4+5+6
21
Exercise
1 + 3 + 5 + 7 + 9 + 11
36
Exercise
2 + 4 + 6 + 8 + 10 + 12
42
Exercise
1 + 2 + 3 + … + 16
136
Sequence
A sequence is a set of
numbers, ordered according
to a distinguishable pattern.
The set can be either finite
or infinite.
conjecture – generalized
statement that seems to
be true
1, 2, 3, 4, 5, 6, 7, …
ellipsis – shows
pattern continues
Each number in a
sequence is called a term.
nth term
an
Position
of the 1
Term
Term
3
2 3
4 5
6 …
6 9 12 15 18 …
Arithmetic Sequence
A sequence of numbers,
each differing by a constant
amount from the preceding
number, is an arithmetic
sequence.
Common Difference
The common difference, d,
is the difference between
successive terms of an
arithmetic sequence.
Arithmetic Sequence
Terms differ by a
constant addend d.
an = an–1 + d
Example 1
Find the value of d for the
sequence 5, 9, 13, 17, 21, 25,
29, 33, …
d=4
Example 1
Find the value of a1 for the
sequence 5, 9, 13, 17, 21, 25,
29, 33, … a1 = 5
Example 1
Find the value of a5 for the
sequence 5, 9, 13, 17, 21, 25,
29, 33, … a5 = 21
Example 1
Find the value of a10 for the
sequence 5, 9, 13, 17, 21, 25,
29, 33, … a10 = 41
Example 2
Write the first six terms of
the sequence in which
a1 = 20 and d = –2.
a1 = 20
a2 = 20 + (–2) = 18
a3 = 18 + (–2) = 16
Example 2
Write the first six terms of
the sequence in which
a1 = 20 and d = –2.
a4 = 16 + (–2) = 14
a5 = 14 + (–2) = 12
a6 = 12 + (–2) = 10
Example 2
Write the first six terms of
the sequence in which
a1 = 20 and d = –2.
The first six terms of
the sequence are 20,
18, 16, 14, 12, and 10.
Example
Is the sequence arithmetic?
4, –8, 16, –32, …
no
Example
Is the sequence arithmetic?
5, 15, 45, 135, …
no
Example
Is the sequence arithmetic?
–4, –2, 0, 2, …
yes
Example
Is the sequence arithmetic?
20, 12, 4, –4, …
yes
Arithmetic Sequence
Terms differ by a
constant addend d.
an = an–1 + d
Recursive Formula
A recursive formula
specifies the step by which
each term of the sequence
is generated from the
preceding term or terms.
Position
of the 1
Term
Term
3
2 3
4 5
6 …
9 15 21 27 33 …
Example 3
Write the first five terms of
the sequence defined by
a1 = 2 and an = an – 1 + 9.
a1 = 2
a2 = a1 + 9 = 2 + 9 = 11
a3 = a2 + 9 = 11 + 9 = 20
Example 3
Write the first five terms of
the sequence defined by
a1 = 2 and an = an – 1 + 9.
a4 = a3 + 9 = 20 + 9 = 29
a5 = a4 + 9 = 29 + 9 = 38
Example 3
Write the first five terms of
the sequence defined by
a1 = 2 and an = an – 1 + 9.
The first five terms of
the sequence are
2, 11, 20, 29, and 38.
Example 4
Write the recursive formula
for the sequence
28, 25, 22, 19, 16, …
a1 = 28
d = –3
an = an – 1 – 3
Example
Write the recursive formula
for the sequence
–4, –2, 0, 2, …
a1 = – 4
an = an – 1 + 2
Example
Write the recursive formula
for the sequence 10, 7, 4, 1, …
an = an – 1 – 3
3, 9 ,15, 21, 27, 33
Add 6 to get the next term.
Position
1
2
3
4
5
6
n
Term
3
3 + 1(6) = 9
3 + 2(6) = 15
3 + 3(6) = 21
3 + 4(6) = 27
3 + 5(6) = 33
3 + (n – 1)6
Explicit Formula
The explicit formula for an
arithmetic sequence is
an = a1 + (n – 1)d, where n is
the position in the sequence
and d is the difference
between terms.
Example 5
The explicit formula for the
sequence 10, 15, 20, 25, 30,
35, … is an = 10 + (n – 1)5.
Find a60.
a60 = 10 + (60 – 1)5
= 10 + (59)5
= 10 + 295
= 305
Example 6
Write the explicit formula for
the sequence
–7, –4, –1, 2, 5, 8, …
a1 = – 7; d = 3
an = a1 + (n – 1)d
= – 7 + (n – 1)3
= – 7 + 3n – 3
= 3n – 10
Example 6
Use the simplified form of the
explicit formula to find a72.
an = 3n – 10
a72 = 3(72) – 10
= 216 – 10
= 206
Example
Write the explicit formula for
the sequence 20, 14, 8, 2, …
an = 20 – (n – 1)6
Example
Write the explicit formula for
the sequence
–10, –7, –4, –1, …
an = –10 + (n – 1)3
Example
Find a9 if an = 6 + (n – 1)2.
22
Example
Find a12 if an = –4 + (n – 1)4.
40
Exercise
If the house numbers on one
side of a street form an
arithmetic sequence where the
first house number is 2, the
second house number is 4, and
the third house number is 6,
how many houses are along
that side of the street if the last
house number is 38? 19
Exercise
The first house number in a
block in 13, and the twelfth
house number is 90.
Assuming the numbers form
an arithmetic sequence, what
is the number of the fourth
house? 34
Exercise
Locust Street has 16 lots for new
houses. If the planners are
laying out the numbers for the
addresses of the new homes on
that street and they are to form
an arithmetic sequence, what is
the largest common difference
that could be used if the
numbers cannot go above 50?
3