SQUARES,TRIANGLE NUMBERS AND FIBONACCI 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 number 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 integer 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 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 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 sequences in the above array? 19 / 30 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 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 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? 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 Factorization 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. 30 / 30.