Perfect Squares

(an  x an+1)

0x1=0 
1x1=1
1x2=2 
2x3=6 
3x5=15 
5x8=40 
8x13=104 
13x21=273 
21x34=714
34x55=1870 

(an  x an+1) – (an  x n-1)

(0x1)-(0x-1)= 0-0=0
(1x1)-(1x0)=1-0=1
(1x2)-(1x1)=2-1=1 
(2x3)-(2x1)=6-2=4 
(3x5)-(3x2)=15-6=9  
(5x8)-(5x3)=40-15=25  
(8x13)-(8x5)=104-40=64  
(13x21)-(13x8)=273-104=169
(21x34)-(21x13)=714-273=441
(34x55)-(34x21)=1870-714=1156 

Pattern

RULE: (anxan+1) – (anxan-1)= A perfect square

Multiples

Key

Multiple of 2
Multiple of 3
Multiple of 5
Multiple of 8

Pattern

Multiple of 2 = Every 3rd Fibonacci Number
Multiple of 3 = Every 4th Fibonacci Number
Multiple of 5 = Every 5th Fibonacci Number
Multiple of 8 = Every 6th Fibonacci Number

RULE: For each successive Fibonacci Number, the number of terms you must wait for there to be a multiple of it rises by 1

Fibonacci Number

1. 1
   
2. 1
   
3. 2
   
4. 3
   
5. 5
   
6. 8/8
   
7. 13
   
8. 21
   
9. 34
   
10. 55
    
11. 89
    
12. 144/144/144
    
13. 233
    
14. 377
    
15. 610/610
    
16. 987
    
17. 1597
    
18. 2584/610
    
19. 4181
    
20. 6765/6765
    
21. 10946
    
22. 17711
    
23. 28657
    
24. 46368/46368/46368
    
25. 75025
    
26. 121393
    
27. 196418
    
28. 317811/317811
    
29. 514229
    
30. 832040/832040/832040