SQUARES,TRIANGLE AND NUMBERS

Todd Cochrane

1 / 30 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

2 / 30 Difference of Consecutive Squares

Geometric Viewpoint: ••••• ••••• ••••• ••••• •••••

Algebraic Viewpoint: n2 − (n − 1)2

3 / 30 Sum of the first n odd numbers

n sum total 1 1 1 2 1 + 3 4 3 1 + 3 + 5 4 1 + 3 + 5 + 7 5 1 + 3 + 5 + 7 + 9 6 1 + 3 + 5 + 7 + 9 + 11

Rule: The sum of the first n odd numbers is ,

4 / 30 Geometric view of sum of odd numbers

••••• ••••• ••••• ••••• •••••

5 / 30 Triangle Numbers

Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, ...

• • • •• • •• • •• ••• •• ••• ••• •••• •••• •••••

Tn = n-th triangle

6 / 30 Differences between consecutive triangle numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, 55,...

Rule: The differences between consecutive triangle numbers are

Tn − Tn−1 =

7 / 30 Triangle Number as a Sum

• •• ••• ••••

Tn =

The n-th triangle number is

8 / 30 Sum of Consecutive Triangle Numbers

•••• •••• •••• ••••

The sum of two consecutive triangle numbers is

Tn−1 + Tn =

Example: T6 + T7 =

Check Answer:

9 / 30 Formula for the n-th Triangle Number Tn

••••• ••••• ••••• •••••

Rule: Tn =

Example: What is the hundredth triangle number?

10 / 30 Sum of the first n natural numbers

We’ve seen two formulas for the n-th triangle number: 1. Tn = 1 + 2 + 3 + ··· + n 1 2. Tn = 2 n(n + 1) Thus we obtain

n(n + 1) 1 + 2 + 3 + ··· + n = 2

Example: Find 1 + 2 + 3 + ··· + 100

11 / 30 Another way to add 1 + 2 + 3 + ··· + 100

12 / 30 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 10 = 9 + 1 2 = 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 is a sum of at most squares. (The same value can be used more than once.)

13 / 30 Sums of Triangle Numbers

Triangle Numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66

1 = 1 10 = 10 2 = 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 most triangle numbers. (The same value can be used more than once.)

14 / 30 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

15 / 30 The Sixth Night: The Fibonacci

“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 .

Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, , ...

Fn = n-th .

Fibonacci Rule: The next term in the Fibonacci sequence is obtained by adding the previous two terms.

Fn+1 = Fn + Fn−1

16 / 30 Fn+1 − Fn = Fn−1

Differences of consecutive Fibonacci numbers

1 1 2 3 5 8 13 21 34 55

17 / 30 Differences of consecutive Fibonacci numbers

1 1 2 3 5 8 13 21 34 55

Fn+1 − Fn = Fn−1

18 / 30 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 Rabbits Number Pairs 1 ◦◦ 1 2 •• 1 3 ◦◦, •• 2 4 ◦◦, ••, •• 3 5 ◦◦, ◦◦, ••, ••, •• 5 6 ◦◦, ◦◦, ◦◦, ••, ••, • • ••, ••, •• 8

Can you see three different Fibonacci in the above array?

19 / 30 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

20 / 30 2 2 Fn + Fn+1 Total 1 + 1 2 1 + 4 5 4 + 9 13 , 9 + 25 34 25 + 64 89 64 + 169 233

2 2 Rule: Fn + Fn+1 = F2n+1

Sum of Consecutive Fibonacci Number Squares

Fn 1 1 2 3 5 8 13 21 34 2 Fn 1 1 4 9 25 64 169 441

21 / 30 2 2 Rule: Fn + Fn+1 = F2n+1

Sum of Consecutive Fibonacci Number Squares

Fn 1 1 2 3 5 8 13 21 34 2 Fn 1 1 4 9 25 64 169 441 2 2 Fn + Fn+1 Total 1 + 1 2 1 + 4 5 4 + 9 13 , 9 + 25 34 25 + 64 89 64 + 169 233

22 / 30 Sum of Consecutive Fibonacci Number Squares

Fn 1 1 2 3 5 8 13 21 34 2 Fn 1 1 4 9 25 64 169 441 2 2 Fn + Fn+1 Total 1 + 1 2 1 + 4 5 4 + 9 13 , 9 + 25 34 25 + 64 89 64 + 169 233

2 2 Rule: Fn + Fn+1 = F2n+1

23 / 30 Partitioning a Rectangle

Draw a rectangle with sides of 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?

24 / 30

2 2 2 F1 + F2 + ··· + Fn = FnFn+1 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!

25 / 30 EXAMPLE 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,...

26 / 30 Rule: The sum of the first n Fibonacci numbers is one less than the (n + 2)-nd Fibonacci number.

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

27 / 30 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.

28 / 30 Prime factors of Fibonacci Numbers

Fn New Prime 2 2 2 3 3 3 5 5 5 8 23 none 13 13 13 34 2 · 17 17 55 5 · 11 11 89 89 89 144 2432 none 233 233 233 377 13 · 29 29 610 2 · 5 · 61 61 987 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. 29 / 30 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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