36
25
4
3
4
49
7
11
Inductive Reasoning
1
0
2
3
3 4 5 6 …
10 21 36 55 …
n
?
… 20
… ?
Linear Sequences
1, 3, 5, 7, 9, …
2, 4, 6, 8, 10, …
3, 8, 13, 18, 23, …
7, 7, 7, 7, 7, …
Sequences
Terms: 4, 5, 6, 7, 8 …
Each number is a term of the sequence. Each term
of the sequence is associated with the counting
numbers. The counting number represent the terms
location: First, second, third, etc.
1 2 3 4 5…
4 5 6 7 6…
1 2 3 4 5…
4 5 6 7 8…
Since each sequence can be thought of and viewed as
an ordered pair, they can be graphed.
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
As you can see the
sequence is a line of
integer values. Hence we
call it a linear sequence.
We can find the succeeding points
by graphing or just visually
recognizing the pattern.
However, graphing is very time consuming.
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
Recognizing the pattern is not efficient for
finding the 50th term because you need to find the
first 49 terms to compute the 50th term.
Therefore, it would be quicker if we
could come up with a simple algebraic rule.
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
G ap 
1 1 1 1
Let’s look at the change in each term or gap
between terms.
Note that the change in y is 1. The change in x is also 1.
T he slope is defined as
change in y
1
 1
change in x 1
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
G ap 
1 1 1 1
T he slope is defined as
change in y
1
 1
change in x 1
Since the sequence is linear it has the following form:
Y = mX + b
m = 1 or the gap between terms.
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
G ap 
1 1 1 1
Since the sequence is linear it has the following form:
Y = mX + b
m = 1 or the gap between terms.
The slope will always be the gap between
terms because the change in x will always be 1.
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 8…
G ap 
Y = (1)X + b
1 1 1 1
To find the value of b, use the first term
and substitute 1 for x and substitute 4
for y.
3=b
4 = (1)(1) + b
4=1+b
The rule is y = x + 3
# of term
Term
Value
x 1 2 3 4 5…
y 4 5 6 7 6…
G ap 
1 1 1 1
The rule is y = x + 3
It works. Look at each term.
5=2+3
7=4+3
6=3+3
8=5+3
1 more time.
That was the first try. Let’s do another.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
Since each sequence can be thought of and
viewed as an ordered pair, they can be graphed.
As you can see the sequence is a
line of integer values. Hence we call it
a linear sequence.
We can find the succeeding points by
graphing or just visually recognizing the
pattern.
However, graphing is very time consuming.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
Recognizing the pattern is not efficient for
finding the 50th term because you need to find the
first 49 terms to compute the 50th term.
Therefore, it would be quicker if we
could come up with a simple algebraic rule.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
Let’s look at the change in each term or gap
between terms.
Note that the change in y is 5. The change in x is also 1.
T he slope is defined as
change in y
change in x

5
1
5
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
T he slope is defined as
change in y
change in x

5
5
1
Since the sequence is linear it has the following form:
Y = mX + b
m = 5 or the gap between terms.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
Since the sequence is linear it has the following form:
Y = mX + b
m = 1 or the gap between terms.
The slope will always be the gap between
terms because the change in x will always be 1.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
Y = 5X + b
6 = 5(1) + b
6=5+b
1=b
5 5 5 5
To find the value of b, use the first term
and substitute 1 for x and substitute 6
for y.
The rule is y = 5x + 1
This is usually done mentally by multiplying
the term # by the gap and figuring out what
else is needed to make the term value.
# of term
Term
Value
x 1 2 3 4 5…
6
11
16
21
26
…
y
G ap 
5 5 5 5
The rule is y = 5x + 1
11 = 5(2) + 1
16 = 5(3) + 1
It works. Look at
each term.
21 = 5(4) + 1
26 = 5(5) + 1
Let’s see if we can do these quickly
# of term
Term
Value
x 1 2 3 4 5… x
9
13
17
21
25
…
4x + 5
y
G ap 
y = mx + b
y = 4x + b
4 4 4 4
9 = 4(1) +b
5=b
y = 4x + 5
Again
# of term
Term
Value
x 1 2 3 4 5… x
5
8
11
14
17
…
3x + 2
y
G ap 
y = mx + b
y = 3x + b
3 3 3 3
5 = 3(1) +b
2=b
y = 3x + 2
And Again
# of term
Term
Value
x 1 2 3 4 5… x
4
11
18
25
32
…
7x - 3
y
G ap 
y = mx + b
y = 7x + b
7 7 7 7
4 = 7(1) +b
-3 = b
y = 7x - 3
One more time !
# of term
Term
Value
x 1 2 3 4 5… x
8
15
22
29
36
…
7x + 1
y
G ap 
y = mx + b
y = 7x + b
7 7 7 7
8 = 7(1) +b
1=b
y = 7x + 1
Linear sequences must be done quickly.
The speed should be almost as fast as you
can write.
The rule is in the form of
where
x+b.
m
x is the number of the term.
Let’s try a few more !
# of term
Term
Value
x 1 2 3 4 5… x
5
7
9
11
13
…
2x + 3
y
G ap 
y = mx + b
y = 2x + b
2 2 2 2
5 = 2(1) +b
3=b
y = 2x + 3
And Another !
# of term
Term
Value
x 1 2 3 4 5… x
8
18
28
38
48
…
10x -2
y
G ap 
y = mx + b
y = 10x + b
10 10 10 10
8 = 10(1) +b
-2 = b
y = 10x - 2
Let’s try tricky ones !
# of term
Term
Value
x 1 2 3 4 5… x
7
7
7
7
7
…
7
y
G ap 
y = mx + b
y = 0x + b
0
0 0 0
7 = 0(1) +b
7=b
y= 7
And Another !
# of term
Term
Value
x 1 2 3 4 5… x
17
14
11
8
5
…
-3x + 20
y
G ap 
y = mx + b
y = -3x + b
-3 -3 -3 -3
17 = -3(1) +b
20 = b
y = -3x + 20
Let’s find the
th
20
x 1 2 3 4 5 6
y 3 8 13 18 23 28
GAP
…
Term
x
3 = 5(1) + b
-2 = + b
20
5x - 2 98
5 55 5
y = 5x + b
…
5(20) - 2
Again
x 1 2 3 4 5 6
y 3 5 7 9 11 13
GAP
…
x
3 = 2(1) + b
1=+b
20
2x + 1 41
2 22 2
y = 2x + b
…
2(20) + 1
And Again
x 1 2 3 4 5 6
y 3 7 11 15 19 23
GAP
…
x
3 = 4(1) + b
-1=+b
20
4x - 1 79
4 44 4
y = 4x + b
…
4(20) - 1
One Last Time
x 1 2 3 4 5 6
y 3 8 13 18 23 28
GAP
…
x
3 = 5(1) + b
-2=+b
20
5x - 2 98
5 55 5
y = 5x + b
…
5(20) - 2
C’est fini.
Good day and good luck.
That’s all folks.
A Senior Citizen Production