factorial

1. n factorial denoted as n!

where 'n' is Whole Number           0, 1, 2, 3, 4, 5, … (and so on)


1!=1*1
2!=1*2
3!=1*2*3
4!=1*2*3*4
.
.
.
n!=1*2*......*(n-1)*n




n!=n*(n-1)!


that is 4!=4*3!


0!  zero factorial =1
n!=n*(n-1)!       .......(1)


1!=1
1!=1*1
1!=1*(1-1)!      from equqtion (1)
1!=1*0!


it implise 0!=1   


number of zero's in n!
number of zero's in n! means that how many time we can divide 10 by n! without  non-zero reminder
example
10!=3628800
10! contain 2 zero
25!=15511210043330985984000000 
25! contain 9 zero's we include only rightmost zeros in this discussion
that is 25! contain only 6 zeros

1. number of zero's in 10^m

          there is 2 zeros in 10!
                           5*2,10
         there is 4 zeros in 20!
                          5*2,15*2,10,20
         there is 7 zeros in  30!
                         5*2,15*2,10,20,30,25*4 gives 2 zeros
         100!contain 24 zeros  
                       10(five)+10(ten)+4(25,50,75,100)


          10 contain 1 zero  m=1
         10!contain 2 zeros 
                      f(1)=2                                 
         100 contain 2 zero m=2
         100!contain  f(2)=f(1)*(10)+2^2m  zeros
                           f(2)=2*10+4
                           f(2)=24
         1000
                m=3
                f(3)=(f(2)*10)+2^3
                f(3)=24*10+8
                f(3)=248


f(m )denoted number of zero's in (10^m)!
m  denote power of 10


f(m)=(f(m-1)*10+2^m)


     


f(m)=2^m+((2^(m-1) )*10)+((2^(m-2) )*100)+......+2*10^(m-1)




                                               

f(m) is number of zero's in   where m is natuaral numbers      1,2,3,4......

example
       number of zero's in 10^6 !

                 





                                                                                                                                                                                                                                                                                                                                    

                          f(0)=0
                          f(1)=2
                          f(2)=24
                          f(3)=248
                          f(4)=2496
                          f(5)=24992
                              .
                              .
                              .
2.       number of zero's in  n!               
                  
              let g(x) is greatest integer function

                  where x is real number
                 that  is  x =5.676
                 then g(x)=5

                example
                                 g(8.0002)=8
                                 g(78.98)=78
                                 g(0.8)    =0
                let F(n)  is a function that denoted number of zero's in n!         

                
           where         d- number digits in n
                              ai - position of bit from right       0,1...  
                                       let   n = 512

                                               a0=2,    a1=1,   a2=5
                         
                             g-  greatest integer function 
                             f- function for find number of zero's in 10^m  !
                                         discuss about f in 1st subtopic
                            F-   F(n)  denoted number of zero's in n! 




          example        n=512
                         F(n) =g(2*(f(0)+0.2))+g(1*(f(1)+0.4))+g(5*(f(2)+0.8))
                                =g(2*(0+0.2))+g(1*(2+0.4))+g(5*(24+0.8))
                                =g(0.4)+g(2.4)+g(g(124)
                                 =0+2+124
                                 =126
                         F(512)=126





                                        

number of digits in n!
     
       To find number of digit in a number find logarith of the number to the base 10


       therefor number of digit in n! is     l( log n!)
       where l is lowest integer function


       that is 
                  l(9.01)=10
                  l(1.6)=2


       n! = 1*2*3*  ......  (n-1)* n
       take logarithm in both side


        log(n!)=log(1) +log (2) +....+log (n-1)+log(n)

         if we use above equation for finding number of digit in n!,
         where n is big number then there is small error in result
          
            so we want new equation for it.
           i try to find it...
           find relation between number of zero's and number of digit's..

answer : progressions

level 1
1)20,24       ap
2)216,1296 gp
3)1/14,1/16 hp
4)375,1875 hp
5)4,2           ap

level 2
1)0.4,0.333        hp     2/1,2/2,2/3,2/4,2/5,2/6  ...
2)-3,3                 gp     r=-1
3)5/4,3/2             ap     d= 1/4
4)7.111,9.481    gp     r=(4/3)
5)-4.8,9.6           gp     r=-2
level 3
1)e
ap  1,2, 3 , 4     d=1
hp  1,2,infinity,-2           1/1,1/0.5,1/0,1/-0.5
gp  1,2,4,8         r=2

sequences

progressions

I)Arithmetic progression (AP) : sequence of numbers such that the difference of any two successive members of the sequence is a constant.

a, a+d, a+2d, a+3d, ....., (a+(n-1)*d)

example:
               3,5,7,9,11,.........,(3+(n-1)*2)
Here,
        2 is common difference (d).
        3 is first number (a).
        n'th number of this sequence is (a+(n-1)*d)


II)Geometric progression (GP) : sequence of numbers such that the quotient of any two successive members of the sequence is a constant.

 a, a*r, a*r², a*r³,.....,a*r^n-1

example:
               4,12,36,108,324,........,4*3^n-1
Here,
        3 is common ratio (r).
        4 is first number (a).
        n'th number of this sequence is (a*r^n-1)

III) Harmonic Progression (HP) : is a progression formed by taking the reciprocals of an arithmetic progression. 

1/a, 1/(a+d), 1/(a+2d),......, 1/(a+(n-1)*d)

example:
               1/3,1/7,1/11,1/15.........[1/(3+(n-1)*4]     in hp   because  3,7,11,15 are in ap.


Find next two numbers..

level 1:

1) 4 , 8 , 12 , 16 , ... , ....
2) 1 , 6 , 36 , ... , ....
3)  1/8 , 1/10 , 1/12 , ... , ....
4) 3 , 15 , 75 , ... , ....
5) 12 , 10 , 8 , 6 , ... , ....

level 2:

1)  2, 1, 0.666, 0.5 , ... , ....
2)  3, -3, 3 , ... , ....
3)  1/2, 3/4, 1 , ... , ....
4)  3, 4, 5.333 , ... , ....
5)  0.6, -1.2, 2.4 , ... , ....

level 3:
1)  1, 2,  , ... , ....
        a)3,4     b)4,8   c) infinity,-2  d) a & b  e) a,b&c


Answers