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Algebraic method of inductive reasoning
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Inductive Reasoning
1 2 3 4 5 6 … n … 20
0 3 10 21 36 55 … ? … ?
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 …
Since each sequence can be thought of and viewed as an ordered pair, they can be graphed.
1 2 3 4 5 …4 5 6 7 8 …
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
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.
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
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.
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
Let’s look at the change in each term or gap between terms.
1 1 1 1
Note that the change in y is 1. The change in x is also 1.
11
1
changein yThe slopeis defined as
changein x
Gap
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
1 1 1 1
11
1
changein yThe slopeis defined as
changein x
Since the sequence is linear it has the following form:Y = mX + b
m = 1 or the gap between terms.
Gap
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
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.
Gap
1 2 3 4 5 …4 5 6 7 8 …
xy
# of term
Term Value
1 1 1 1
Y = (1)X + b To find the value of b, use the first term and substitute 1 for x and substitute 4 for y.
4 = (1)(1) + b
4 = 1 + b
3 = b
The rule is y = x + 3
Gap
1 2 3 4 5 …4 5 6 7 6 …
xy
# of term
Term Value
1 1 1 1
It works. Look at each term.The rule is y = x + 3
5 = 2 + 3
6 = 3 + 3
7 = 4 + 3
8 = 5 + 3
Gap
That was the first try. Let’s do another.
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
1 more time.
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.
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.
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
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.
55
1
changein yThe slopeis defined as
changein x
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
55
1
changein yThe slopeis defined as
changein x
Since the sequence is linear it has the following form:Y = mX + b
m = 5 or the gap between terms.
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
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.
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
Y = 5X + b To find the value of b, use the first term and substitute 1 for x and substitute 6 for y.
6 = 5(1) + b
6 = 5 + b
1 = b
The rule is y = 5x + 1This is usually done mentally by multiplying the term # by the gap and figuring out what else is needed to make the term value.
1 2 3 4 5 …6 11 16 21 26 …
xy
# of term
Term Value
5 5 5 5Gap
The rule is y = 5x + 1
11 = 5(2) + 1
It works. Look at each term.
16 = 5(3) + 1
21 = 5(4) + 1
26 = 5(5) + 1
Let’s see if we can do these quickly
1 2 3 4 5 … x9 13 17 21 25 …
xy
# of term
Term Value
4 4 4 4Gap
y = mx + b y = 4x + b
9 = 4(1) +b5 = by = 4x + 5
4x + 5
Again
1 2 3 4 5 … x5 8 11 14 17 …
xy
# of term
Term Value
3 3 3 3Gap
y = mx + b y = 3x + b
5 = 3(1) +b2 = by = 3x + 2
3x + 2
And Again
1 2 3 4 5 … x4 11 18 25 32 …
xy
# of term
Term Value
7 7 7 7Gap
y = mx + b y = 7x + b
4 = 7(1) +b-3 = by = 7x - 3
7x - 3
One more time !
1 2 3 4 5 … x8 15 22 29 36 …
xy
# of term
Term Value
7 7 7 7Gap
y = mx + b y = 7x + b
8 = 7(1) +b1 = by = 7x + 1
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 mx + b .
where x is the number of the term.
Let’s try a few more !
1 2 3 4 5 … x5 7 9 11 13 …
xy
# of term
Term Value
2 2 2 2Gap
y = mx + b y = 2x + b
5 = 2(1) +b3 = by = 2x + 3
2x + 3
And Another !
1 2 3 4 5 … x8 18 28 38 48 …
xy
# of term
Term Value
10 10 10 10Gap
y = mx + b y = 10x + b
8 = 10(1) +b-2 = b
y = 10x - 2
10x -2
Let’s try tricky ones !
1 2 3 4 5 … x7 7 7 7 7 …
xy
# of term
Term Value
0 0 0 0Gap
y = mx + b y = 0x + b
7 = 0(1) +b7 = b
y = 7
7
And Another !
1 2 3 4 5 … x17 14 11 8 5 …
xy
# of term
Term Value
-3 -3 -3 -3Gap
y = mx + b y = -3x + b
17 = -3(1) +b 20 = b
y = -3x + 20
-3x + 20
Let’s find the 20th Termx 1 2 3 4 5 6
… x … 20
y
GAP
3 13 18 23 288
5 5 5 5 y = 5x + b
3 = 5(1) + b
-2 = + b
5x - 2
5(20) - 2
98
x 1 2 3 4 5 6 … x
… 20
y
GAP
3 7 9 11 135
2 2 2 2
Again
y = 2x + b
3 = 2(1) + b
1 = + b
2x + 1
2(20) + 1
41
x 1 2 3 4 5 6 … x
… 20
y
GAP
3 11 15 19 237
4 4 4 4
And Again
y = 4x + b
3 = 4(1) + b
- 1 = + b
4x - 1
4(20) - 1
79
One Last Time
x 1 2 3 4 5 6 … x
… 20
y
GAP
3 13 18 23 288
5 5 5 5 y = 5x + b
3 = 5(1) + b
- 2 = + b
5x - 2
5(20) - 2
98
C’est fini.Good day and good luck.
A Senior Citizen Production
That’s all folks.
