1 digit: only 1 digit.
2 digits: each digit can be 0 to 6, with 7 digits in total.
3 digits:
When the first digit is 1, the number of digits can range from 0 to 6 or 7.
When the first digit is 2, the number can range from 0 to 5 or 6.
. . .
When the first digit is 7, the number of digits can range from 0 to 0, 1.
4 digits:
When the current two digits are 10, the range of digits can be 0 to 6 or 7.
When the current two digits are 1 1, the range of digits can be 0 to 5 or 6.
. . .
When the current two digits are 16, the range of digits can be 0 to 0 and 1.
When the current two digits are 20, the range of digits can be 0 to 5 or 6.
When the current two digits are 2 1, the range of digits can be 0 to 4 or 5.
. . .
When the current two digits are 25, the range of digits can be 0 to 0, 1.
. . .
When the current two digits are 70, the range of digits can be 0 to 0, 1.
As you can see, when there are three numbers, there are the following series:
C3: 1 2 ... Seven
When the number is 4, there are the following series:
C4: 1...( 1+2+...+7)
The general formula of sequence C4 is the sum of the first n terms of sequence C3.
Similarly, the 5-digit sequence C5 will be the sum of the first n terms of the sequence C4.
The sum of the first n terms of sequence C3 is: S3 = n (n+ 1)/2, (n = 1, 2, ..., 7).
The sum of the first n terms of the sequence C4 is:
S4 =[ 1 * 2+2 * 3]...+n(n+ 1)]/2
=[( 1^2+2^2+...+n^2)+( 1+2+...+n)]/2
=[n(n+ 1)(2n+ 1)/6+n(n+ 1)/2]/2
=n(n+ 1)(n+2)/6，(n= 1，2，...,7)
The sum of the first n terms of sequence C5 is:
S5=[ 1*2*3+2*3*4+...+n(n+ 1)(n+2)]/6
=[2*(2^2- 1)+3*(3^2- 1)+...+(n+ 1)*((n+ 1)^2- 1)]/6
=[(2^3+3^3+...+(n+ 1)^3)-(2+3+...+(n+ 1))]/6
=[( 1+2+...+(n+ 1))^2- 1-(2+3+...+(n+ 1))]/6
=n(n+ 1)(n+2)(n+3)/24，(n= 1，2，...,7)
Looking at this law, we can know that the sum of the first n terms of C6 sequence is:
S6 = n(n+ 1)(n+2)(n+3)(n+4)/ 120，(n= 1，2，...,7)
To sum up, the year 2005 is n =1+7+2 * (1+2+...+7)+1= 65.
Now look for 5N=5*65=325.
S 1= 1
S2=7
When n=7 and S3=28, it is obvious that item 5 is not a 3-digit number.
When n=7 and S4=84, it is obvious that item 5 is not a 4-digit number.
When n=7, S5=2 10, s1+S2+...+S5 = 330 > 325,
So the fifth item is five digits, the sixth from the bottom, which is 52000.