Transcript Lesson 5 (3rd 6 Weeks) TEKS 6.4 A/B

Arithmetic
Sequences
Lesson 5 (3rd 6 Weeks)
TEKS 6.4 A/B
Sequence
• A set of numbers written in a particular
order.
– For Example: 6, 10, 14, 18 is a
sequence of four numbers. The number
6 is the first term in the sequence, 10 is
the second term, 14 is the third term,
and 18 is the fourth term.
Arithmetic Sequence
• A sequence of numbers where the
difference between the successive terms
is constant.
– For Example: The first five terms of an
arithmetic sequence are 3, 6, 9, 12,
15… The number 3 is the first term in
the sequence, 6 is the second term, 9 is
the third term, 12 is the fourth term, and
15 is the fifth term.
• The common difference in an arithmetic
sequence can be identified by finding the
difference between the terms in the
sequence.
+3 +3 +3 +3
3, 6, 9, 12, 15,…
+3 +3 +3 +3
3, 6, 9, 12, 15,…
• In the sequence 3, 6, 9, 12, 15,… the
common difference is 3.
Follow these guidelines to find a rule or
expression that can be used to find the
nth term in an arithmetic sequence:
1. Use the common difference to find a
pattern that shows the relationship
between the term’s position number and
the value of the term.
2. Multiply the common difference and the
position number.
3. Adjust by adding or subtracting to get the
value of the term needed.
4. State the pattern as a rule.
5. Check to see whether the rule works for
the next two terms in the sequence.
6. Represent the rule as an algebraic
expression.
Example 1:
Position #
1
2
3
4
5
n
Value of the
Term (VOT)
3
6
9
12
15
?
Position #
1
2
3
4
5
n
1x3=
2x3=
3x3=
4x3=
5x3=
nx3
Value of the Term
(VOT)
3
+3
6
+3
9
+3
12
+3
15
?
• The common difference of the “Value
of the Term” is 3.
• Multiply the position number by the
common difference.
Position #
1
2
3
1•3=3
2•3=6
3•3=9
4
5
N
4 • 3 = 12
5 • 3 = 15
n•3
Value of the Term
(VOT)
3
6
9
12
15
?
• The pattern is “multiply the position
number by 3 to get the value of the term.”
• Written as an expression: n x 3 or 3n
Example:
Notice the
numbers in
the x column
are
successive.
x
1
2
3
4
n
y
2
5
8
11
+3
+3
+3
x
1x 3 =
2
3
4
n
y
2
5
8
11
+3
+3
+3
• In the sequence 2, 5, 8, 11 …, the
common difference is 3.
• Multiply the common difference times the
x-value. 1 x 3 = 3
• The first y-value is 2, not 3, therefore you
must add or subtract from 3 to find the yvalue (adjust). 3 -1 = 2
x
y
1x 3 – 1 = 2
2x 3 – 1 = 5
3x 3 – 1 = 8
4 x 3 – 1 = 11
nx 3 – 1 =
• Maybe each y-value in this sequence is
equal to 3 times its x-value an subtract 1.
x • 3 - 1 = y or 3x – 1 = y
• Check to see whether the rule works for
the next two terms in the sequence.
2 x 3 -1 = 5
3x3–1=8
x
1x 3 – 1 =
2x 3 – 1 =
3x 3 – 1 =
4x 3 – 1 =
n x3–1=
y
2
5
8
11
• Represent the rule as an algebraic
expression.
3n - 1
Example:
x
y
1
7
3
15
5
23
+4
6
27
• Notice that the x-values are not
successive until you get to the values of 5
and 6. You can only look for the common
difference when the terms are successive.
x
y
1
7
x
4
+
=
3
=
4x + 3
3 x 4 + 3 = 15
5 x 4 + 3 =23
+4
6 x 4 + 3 = 27
• Multiply 4 by the x values. Notice that
when you do, you don’t get the y-values.
• So you must add or subtract to get the yvalues.
• Check to see if the rule works. Then write
it algebraic.