Download Squares, Triangle Numbers and Fibonacci Numbers

S QUARES , T RIANGLE N UMBERS
AND F IBONACCI N UMBERS
Todd Cochrane
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The Fifth Night: Squares and Triangle Numbers
Squares: 12 , 22 , 32 , 42 , 52 , . . . = 1, 4, 9, 16, 25, 36, 49, . . .
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Differences of Consecutive Squares:
1,
4,
9,
16,
25,
36,
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49,
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64,
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81,
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Rule: The differences of consecutive squares are the
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Difference of Consecutive Squares
Geometric Viewpoint:
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Algebraic Viewpoint:
n2 − (n − 1)2
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Sum of the first n odd numbers
n
1
2
3
4
5
6
sum
total
1
1
1+3
4
1+3+5
1+3+5+7
1+3+5+7+9
1 + 3 + 5 + 7 + 9 + 11
Rule: The sum of the first n odd numbers is
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Geometric view of sum of odd numbers
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Triangle Numbers
Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, ...
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Tn = n-th triangle number
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Differences between consecutive triangle numbers
1,
3,
6,
10,
15,
21,
28,
36,
45,
5
Rule: The differences between consecutive triangle
numbers are
Tn − Tn−1 =
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Triangle Number as a Sum
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Tn =
The n-th triangle number is
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Sum of Consecutive Triangle Numbers
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The sum of two consecutive triangle numbers is
Tn−1 + Tn =
Example: T6 + T7 =
Check Answer:
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Formula for the n-th Triangle Number Tn
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Rule: Tn =
Example: What is the hundredth triangle number?
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Sum of the first n natural numbers
We’ve seen two formulas for the n-th triangle number:
1. Tn = 1 + 2 + 3 + · · · + n
2. Tn = 12 n(n + 1)
Thus we obtain
1 + 2 + 3 + ··· + n =
n(n + 1)
2
Example: Find 1 + 2 + 3 + · · · + 100
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Another way to add 1 + 2 + 3 + · · · + 100
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Sums of Squares
Represent n as a sum of squares using as few squares as
possible.
Squares: 1,4,9,16,25,36,49,64,81,...
1=1
2=1+1
3=1+1+1
4=4
5=4+1
6=4+1+1
7=4+1+1+1
8=4+4
9=9
10 = 9 + 1
20 =
30 =
40 =
50 =
60 =
70 =
80 =
90 =
Fact: Every positive integer is a sum of at most
squares. (The same value can be used more than
once.)
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Sums of Triangle Numbers
Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66
1=1
2=1+1
3=3
4=3+1
5=3+1+1
6=3+3
7=3+3+1
8=6+1+1
9=6+3
10 = 10
20 =
30 =
40 =
50 =
60 =
70 =
80 =
90 =
Fact: Every positive integer is a sum of at most
triangle numbers. (The same value can be used more
than once.)
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Further Properties of Squares and Triangle Numbers
• There are infinitely many triangle numbers that are squares,
T1 = 1, T8 = 36, T49 = 1225,...
• A positive integer n is a triangle number if and only if 8n + 1 is
a square.
• The sum of the reciprocals of all triangle numbers is
1+
1 1
1
1
1
1
+ +
+
+
+
+ ··· =
3 6 10 15 21 28
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The Sixth Night: The Fibonacci Sequence
“Lots of number devils in Number Heaven. The bosses do
nothing but sit and think. One boss is named Bonacci (for
Fibonacci)."
Fibonacci lived 1170-1250. Fibonacci sequence appears
earlier in Indian mathematics.
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,
, ...
Fn = n-th Fibonacci Number.
Fibonacci Rule: The next term in the Fibonacci sequence
is obtained by adding the previous two terms.
Fn+1 = Fn + Fn−1
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Differences of consecutive Fibonacci numbers
1
1
2
3
5
8
13
21
34
55
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Differences of consecutive Fibonacci numbers
1
1
2
3
5
8
13
21
34
55
Fn+1 − Fn = Fn−1
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Snow Rabbits
Reproduction Rule:
I. Start with a pair of newborn snow rabbits (one male, one
female): ◦◦
II. After one month snow rabbits turn brown: ••
III. After another month they have a pair of babies (one male,
one female) and then continue to have a pair each month
thereafter.
Month
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2
3
4
5
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Rabbits
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◦◦, ••, ••
◦◦, ◦◦, ••, ••, ••
◦◦, ◦◦, ◦◦, ••, ••, • • ••, ••, ••
Number Pairs
1
1
2
3
5
8
Can you see three different Fibonacci sequences in the above
array?
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Tree Branching:
Branching Rule:
I. Start with a stem (with no branches).
II. After two years of growth a new branch is formed, and then a
new branch is formed each year thereafter.
III. Each new branch follows the same rule as the original stem.
year 6
year 5
year 4
year 3
year 2
year 1
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34
Fn2 1 1 4 9 25 64 169 441
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34
Fn2 1 1 4 9 25 64 169 441
2
Fn2 + Fn+1
Total
1+1
2
1+4
5
13 ,
4+9
9 + 25
34
25 + 64
89
64 + 169
233
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34
Fn2 1 1 4 9 25 64 169 441
2
Fn2 + Fn+1
Total
1+1
2
1+4
5
13 ,
4+9
9 + 25
34
25 + 64
89
64 + 169
233
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Rule: Fn2 + Fn+1
= F2n+1
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Partitioning a Rectangle
Draw a rectangle with sides of lengths F3 , F4 and partition it
into squares with side lengths F1 , F2 and F3 .
Do the same thing for F4 , F5 .
What formula do you discover?
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Breaking up a number as a sum of Fibonacci numbers
Fact: Every positive integer can be expressed uniquely
as a sum of one or more distinct Fibonacci numbers no
two of which are consecutive.
Compare this concept with factoring numbers. What is the
difference?
Procedure: Start with the biggest Fibonacci number less than
or equal to the given number, see what’s left over, and repeat!
It’s a lot easier than factoring!
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E XAMPLE 1
Express 135 and 150 as a sum of distinct Fibonacci numbers,
no two consecutive: 1,2,3,5,8,13,21,34,55,89,144,...
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Sum of Fibonacci numbers
The Fibonaccis: 1,1,2,3,5,8,13,21,34,55,89,144,...
F1 + F2 + F3 + · · · + Fn
Total
1
1
1+1
2
1+1+2
4
1+1+2+3
7
1+1+2+3+5
12
1+1+2+3+5+8
20
1 + 1 + 2 + 3 + 5 + 8 + 13
33
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21
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Sum of Fibonacci numbers
The Fibonaccis: 1,1,2,3,5,8,13,21,34,55,89,144,...
F1 + F2 + F3 + · · · + Fn
Total
1
1
1+1
2
1+1+2
4
1+1+2+3
7
1+1+2+3+5
12
1+1+2+3+5+8
20
1 + 1 + 2 + 3 + 5 + 8 + 13
33
1 + 1 + 2 + 3 + 5 + 8 + 13 + 21
Rule: The sum of the first n Fibonacci numbers is one
less than the (n + 2)-nd Fibonacci number.
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Prime factors of Fibonacci Numbers
Fn Factorization New Prime
2
2
2
3
3
3
5
5
5
3
2
none
8
13
13
13
2 · 17
17
34
55
5 · 11
11
89
89
89
4
2
144
2 3
none
233
233
233
377
13 · 29
29
2 · 5 · 61
61
610
987
3 · 7 · 47
7, 47
Fact: Every Fibonacci number has a prime factor that
is not a factor of any earlier Fibonacci number, except
1,8 and 144.
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Further remarks
• The only square Fibonacci numbers are 0, 1 and 144.
• The sum of the first n even numbered Fibonacci numbers is
one less than the next Fibonacci number.
• The sum of the first n odd numbered Fibonacci numbers is
the next Fibonacci number.
• If d is a factor of n, then Fd is a factor of Fn .
Example: 6 is a factor of 12. F6 = 8, F12 = 144. 8 is a factor of
144.
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