The Fifth Night: Squares and Triangle Numbers
Squares: 12,22,32,42,52, . . . = 1,4,9,16,25,36,49, . . .
• • •• •
• • •• • •• • •
• • • •• • • •• • • •• • • •
• • • • •• • • • •• • • • •• • • • •• • • • •
Differences of Consecutive Squares:1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Rule: The differences of consecutive squares are the
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Difference of Consecutive Squares
Geometric Viewpoint:
• • • • •• • • • •• • • • •• • • • •• • • • •
Algebraic Viewpoint:
n2 − (n − 1)2
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Sum of the first n odd numbers
n sum total1 1 12 1 + 3 43 1 + 3 + 54 1 + 3 + 5 + 75 1 + 3 + 5 + 7 + 96 1 + 3 + 5 + 7 + 9 + 11
Rule: The sum of the first n odd numbers is ,
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Triangle Numbers
Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, ...
• •• •
•• •• • •
•• •• • •• • • •
•• •• • •• • • •• • • • •
Tn = n-th triangle number
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Differences between consecutive triangle numbers
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, . . .
Rule: The differences between consecutive trianglenumbers are
Tn − Tn−1 =
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Sum of Consecutive Triangle Numbers
• • • •• • • •• • • •• • • •
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
• • • • •• • • • •• • • • •• • • • •
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 = 12n(n + 1)
Thus we obtain
1 + 2 + 3 + · · ·+ n =n(n + 1)
2
Example: Find 1 + 2 + 3 + · · ·+ 100
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Sums of Squares
Represent n as a sum of squares using as few squares aspossible.
Squares: 1,4,9,16,25,36,49,64,81,...
1 = 1 10 = 9 + 12 = 1 + 1 20 =3 = 1 + 1 + 1 30 =4 = 4 40 =5 = 4 + 1 50 =6 = 4 + 1 + 1 60 =7 = 4 + 1 + 1 + 1 70 =8 = 4 + 4 80 =9 = 9 90 =
Fact: Every positive integer is a sum of at mostsquares. (The same value can be used more thanonce.)
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Sums of Triangle Numbers
Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66
1 = 1 10 = 102 = 1 + 1 20 =3 = 3 30 =4 = 3 + 1 40 =5 = 3 + 1 + 1 50 =6 = 3 + 3 60 =7 = 3 + 3 + 1 70 =8 = 6 + 1 + 1 80 =9 = 6 + 3 90 =
Fact: Every positive integer is a sum of at mosttriangle numbers. (The same value can be used morethan 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 isa square.
• The sum of the reciprocals of all triangle numbers is
1 +13+
16+
110
+115
+121
+128
+ · · · =
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The Sixth Night: The Fibonacci Sequence
“Lots of number devils in Number Heaven. The bosses donothing but sit and think. One boss is named Bonacci (forFibonacci)."
Fibonacci lived 1170-1250. Fibonacci sequence appearsearlier 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 sequenceis obtained by adding the previous two terms.
Fn+1 = Fn + Fn−1
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Snow RabbitsReproduction Rule:
I. Start with a pair of newborn snow rabbits (one male, onefemale): ◦◦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 monththereafter.
Month Rabbits Number Pairs1 ◦◦ 12 •• 13 ◦◦, •• 24 ◦◦, ••, •• 35 ◦◦, ◦◦, ••, ••, •• 56 ◦◦, ◦◦, ◦◦, ••, ••, • • ••, ••, •• 8
Can you see three different Fibonacci sequences in the abovearray?
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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 anew branch is formed each year thereafter.III. Each new branch follows the same rule as the original stem.
year 6year 5year 4year 3year 2year 1
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34F 2
n 1 1 4 9 25 64 169 441
F 2n + F 2
n+1 Total1 + 1 21 + 4 54 + 9 139 + 25 3425 + 64 8964 + 169 233
,
Rule: F 2n + F 2
n+1 = F2n+1
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34F 2
n 1 1 4 9 25 64 169 441
F 2n + F 2
n+1 Total1 + 1 21 + 4 54 + 9 139 + 25 3425 + 64 8964 + 169 233
,
Rule: F 2n + F 2
n+1 = F2n+1
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Sum of Consecutive Fibonacci Number Squares
Fn 1 1 2 3 5 8 13 21 34F 2
n 1 1 4 9 25 64 169 441
F 2n + F 2
n+1 Total1 + 1 21 + 4 54 + 9 139 + 25 3425 + 64 8964 + 169 233
,
Rule: F 2n + F 2
n+1 = F2n+1
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Partitioning a Rectangle
Draw a rectangle with sides of lengths F3, F4 and partition itinto squares with side lengths F1,F2 and F3.
Do the same thing for F4,F5.
What formula do you discover?
F 21 + F 2
2 + · · ·+ F 2n = FnFn+1
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Breaking up a number as a sum of Fibonacci numbers
Fact: Every positive integer can be expressed uniquelyas a sum of one or more distinct Fibonacci numbers notwo of which are consecutive.
Compare this concept with factoring numbers. What is thedifference?
Procedure: Start with the biggest Fibonacci number less thanor equal to the given number, see what’s left over, and repeat!
It’s a lot easier than factoring!
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EXAMPLE 1Express 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 Total1 11 + 1 21 + 1 + 2 41 + 1 + 2 + 3 71 + 1 + 2 + 3 + 5 121 + 1 + 2 + 3 + 5 + 8 201 + 1 + 2 + 3 + 5 + 8 + 13 331 + 1 + 2 + 3 + 5 + 8 + 13 + 21
Rule: The sum of the first n Fibonacci numbers is oneless than the (n + 2)-nd Fibonacci number.
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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 Total1 11 + 1 21 + 1 + 2 41 + 1 + 2 + 3 71 + 1 + 2 + 3 + 5 121 + 1 + 2 + 3 + 5 + 8 201 + 1 + 2 + 3 + 5 + 8 + 13 331 + 1 + 2 + 3 + 5 + 8 + 13 + 21
Rule: The sum of the first n Fibonacci numbers is oneless than the (n + 2)-nd Fibonacci number.
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Prime factors of Fibonacci Numbers
Fn Factorization New Prime2 2 23 3 35 5 58 23 none
13 13 1334 2 · 17 1755 5 · 11 1189 89 89144 2432 none233 233 233377 13 · 29 29610 2 · 5 · 61 61987 3 · 7 · 47 7,47
Fact: Every Fibonacci number has a prime factor thatis not a factor of any earlier Fibonacci number, except1,8 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 isone less than the next Fibonacci number.
• The sum of the first n odd numbered Fibonacci numbers isthe 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 of144.
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