SQUARES, TRIANGLE NUMBERSAND FIBONACCI NUMBERS

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, 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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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, 55, . . .

Rule: The differences between consecutive trianglenumbers are

Tn − Tn−1 =

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Triangle Number as a Sum

•• •• • •• • • •

Tn =

The n-th triangle number is

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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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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 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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Differences of consecutive Fibonacci numbers

1 1 2 3 5 8 13 21 34 55

Fn+1 − Fn = Fn−1

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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 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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