Download 2 + 3

Problem:
Add the first 100 counting numbers
together.
1 + 2 + …+ 99 + 100
We shall see if we can find a fast way of doing this
problem.
Designed by David Jay Hebert, PhD
A pattern of numbers in a
particular order is called a
number sequence, and the
individual numbers in the
sequence are called terms.
Designed by David Jay Hebert, PhD
Each number or term in
the sequence is associated
with a position, which is
also a number.
Designed by David Jay Hebert, PhD
Examples of Sequences
1,2,3,4,5,….
1,1,2,3,5,8,…
1,2,4,8,16,32,…
2,1,7,3,9,3,…
The dots at the end of the sequence indicate that the
sequence continues without end. Note not all sequences
have a pattern.
Designed by David Jay Hebert, PhD
Arithmetic Sequences
The sequence
2, 5, 8, 11, 14,…
Has first differences of 3 all the time, this
makes this sequence arithmetic. If a
sequence is arithmetic the first differences
must be the same.
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First Differences
First differences are the differences
found by subtracting two consecutive
numbers in a sequence.
Ex.
2, 5, 8, 11, 14,….
Designed by David Jay Hebert, PhD
First Differences
First differences are the differences
found by subtracting two consecutive
numbers in a sequence.
Ex.
2, 5, 8, 11, 14,….
3
5–2=3
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First Differences
First differences are the differences
found by subtracting two consecutive
numbers in a sequence.
Ex.
2, 5, 8, 11, 14,….
3 3
8–5=3
Designed by David Jay Hebert, PhD
First Differences
First differences are the differences
found by subtracting two consecutive
numbers in a sequence.
Ex.
2, 5, 8, 11, 14,….
3 3 3
11 - 8 = 3
Designed by David Jay Hebert, PhD
First Differences
First differences are the differences
found by subtracting two consecutive
numbers in a sequence.
Ex.
2, 5, 8, 11, 14,….
3 3 3
3
Since all the first differences are the
same this must be an arithmetic
sequence Designed by David Jay Hebert, PhD
Position Numbers
Each term in the sequence is paired
with a position number
2, 5, 8, 11, 14,…
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Arithmetic Sequences
2, 5, 8, 11, 14,…
(recall the first difference is 3)
5=2+3
Designed by David Jay Hebert, PhD
Arithmetic Sequences
2, 5, 8, 11, 14,…
(recall the first difference is 3)
5=2+3
8=5+3=2+3+3
(the underlined part is the previous term)
Designed by David Jay Hebert, PhD
Arithmetic Sequences
2, 5, 8, 11, 14,…
(recall the first difference is 3)
5=2+3
8=5+3=2+3+3
11 = 8 + 3 = 2 + 3 + 3 + 3
(the underlined part is the previous term)
Designed by David Jay Hebert, PhD
Arithmetic Sequences
2, 5, 8, 11, 14,…
(recall the first difference is 3)
5=2+3
8=5+3=2+3+3
11 = 8 + 3 = 2 + 3 + 3 + 3
14 = 11 + 3 = 2 + 3 + 3 + 3 + 3
(the underlined part is the previous term)
Designed by David Jay Hebert, PhD
Arithmetic Sequence:
Predictability
The question is how many times does one
move from the first term in a sequence.
Ex.
2, 5, 8, 11, 14,….
Take one step from 2 to get to 5
Take two steps from 2 to get to 8
Take three steps from 2 to get to 11
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Arithmetic Sequences
• Have a constant first difference
• Are predictable
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Predictability
If the sequence in question is arithmetic the
position numbers will begin with a 0th
position.
i.e.,
2, 5, 8, 11, 14, …
Designed by David Jay Hebert, PhD
Predictability
If the sequence in question is arithmetic the
position numbers will begin with a 0th
position.
i.e.,
2, 5, 8, 11, 14, …
2 is in the 0th position. Zero 3’s were added to
2 to get to 2. But 2 is the 1st term in the
sequence.
Designed by David Jay Hebert, PhD
Predictability
If the sequence in question is arithmetic the
position numbers will begin with a 0th
position.
i.e.,
2, 5, 8, 11, 14, …
5 is in the 1st position. One 3 was added to 2
to get to 5. But 5 is the 2nd term in the
sequence.
Designed by David Jay Hebert, PhD
Predictability
If the sequence in question is arithmetic the
position numbers will begin with a 0th
position.
i.e.,
2, 5, 8, 11, 14, …
8 is in the 2nd position. Two 3’s were added
to 2 to get to 8. But 8 is the 3rd term in the
sequence.
Designed by David Jay Hebert, PhD
Predictability
If the sequence in question is arithmetic the
position numbers will begin with a 0th
position.
i.e.,
2, 5, 8, 11, 14, …
11 is in the 3rd position. Three 3’s were added
to 2 to get to 11. But 11is the 4th term in the
sequence.
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Predictability
If we wanted to know the term in the 22nd
position, we would add 22 threes to 2 to get
the result, i.e.,
22(3) +2 = 66 + 2 = 68.
68 is in the 22nd position, but it is the 23rd term
in the sequence.
Designed by David Jay Hebert, PhD
Predictability
Mathematics is the study of patterns, and
therefore we must look for a pattern.
The position number is the same as the
number of first differences that must be added
to the first term.
This leads me to believe that the following
formula might work.
Term = (Position Number)(Step Size) + Initial Step
Designed by David Jay Hebert, PhD
Arithmetic Sequences
Start with a sequence
-2, 3, 8, 13, 18, 23,…
Find the step size (first difference).
3 – (-2) = 5
Find the initial step (term in 0th position)
-2
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Arithmetic Sequences
Term = (position)(step size) + initial step
From above
Term= (position)(5) + -2
This is a relationship between the position of a
term and the term in a position.
Designed by David Jay Hebert, PhD
Arithmetic Sequences
Term = (position)(step size) + initial step
From above
Term= (position)(5) + -2
This is a relationship between the position of a
term and the term in a position.
What is the 45th term?
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Arithmetic Sequence
Term = (position)(5) – 2
Term = (45)(5) – 2
Term = 210 – 2
Term = 208
By knowing the position we are
able to determine the term, what
about the other way around.
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Arithmetic Sequence
What if we know the term but not the
position, can the position be determined?
Suppose that 183 is a number in the
sequence
–2, 3, 8, 13, 18,…
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Arithmetic Sequence
The relationship
Term = (Position)(Step Size) + Initial Step
Replace with known values
183 = (Position)(5) – 2
37 = Position
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New Problem:
Find the sum of the sequence:
-2, 3, 8, 13, …,603, 608, 613
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New Problem:
S = -2 + 3 + 8 + … + 603 + 608 + 613
Is the same as
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8
+ 3 + (-2)
Add down in columns defined by
the plus signs.
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
If we knew the number of terms in the
original sequence we could answer the
question 611+ 611 + … + 611 + 611.
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
The sequence –2, 3, 8, 13,…
Is arithmetic therefore we have
predictability.
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
613 = (Position)(5) – 2
123 = Position, which means there are 124
terms in the sequence.
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
There are 124 - 611’s in the sequence
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
2S = (611)(124)
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
2S = (611)(124)
2S = 75764
Designed by David Jay Hebert, PhD
New Problem:
Notice the following
S = -2
+ 3 + 8 + … + 603 + 608 + 613
S = 613 + 608 + 603 + … + 8 + 3 + (-2)
2S = 611 + 611+ 611 + … + 611 + 611 + 611
2S = (611)(124)
2S = 75764
S = 37882
Designed by David Jay Hebert, PhD
New Problem:
Let us find the sum of the sequence
5, 9, 13, … , 365, 369, 373
First we must find out how many terms
are in the sequence. In order to do this
we must determine the step size and
initial step of the sequence.
Designed by David Jay Hebert, PhD
New Problem:
Let us find the sum of the sequence
5, 9, 13, … , 365, 369, 373
Step Size = 4
Initial Step = 5
Hence
Term = 4(Position) + 5
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New Problem:
Let us find the sum of the sequence
5, 9, 13, … , 365, 369, 373
Term = 4(Position) + 5
373 = 4(Position) + 5
92 = Position
We will need this information later.
Designed by David Jay Hebert, PhD
Step 1: Write down the sum you wish to determine
S=5 +9
+ 13 + … + 365 + 369 + 373
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Step 2: reverse the order of the sum and write it under the first
sum.
S=5 +9
+ 13 + … + 365 + 369 + 373
S = 373 + 369 + 365 + …+ 13 + 9 + 5
Designed by David Jay Hebert, PhD
Step 3: add down in columns determined by the plus signs in
the problem.
S=5 +9
+ 13 + … + 365 + 369 + 373
S = 373 + 369 + 365 + …+ 13 + 9 + 5
2S = 378 + 378 + 378+ … + 378 + 378 + 378
Notice that all the columns have the same
sum, if only we knew how many they were!
Designed by David Jay Hebert, PhD
Step 4: recall we had predictability with arithmetic sequences
and we already determined that 373 was in position 92 of the
original sequence. Therefore 93 terms in the sequence.
S=5 +9
+ 13 + … + 365 + 369 + 373
S = 373 + 369 + 365 + …+ 13 + 9 + 5
2S = 378 + 378 + 378+ …+ 378 + 378 + 378
Designed by David Jay Hebert, PhD
Step 5: replace the long sum with the associated
multiplication problem.
S=5 +9
+ 13 + … + 365 + 369 + 373
S = 373 + 369 + 365 + …+ 13 + 9 + 5
2S = 378 + 378 + 378+ …+ 378 + 378 + 378
2S = 378(93)
Designed by David Jay Hebert, PhD
Step 6: solve for S and be done with the problem.
S=5 +9
+ 13 + … + 365 + 369 + 373
S = 373 + 369 + 365 + …+ 13 + 9 + 5
2S = 378 + 378 + 378+ …+ 378 + 378 + 378
2S = 378(93)
2S = 35154
S = 17577
Designed by David Jay Hebert, PhD