Find the first number in the following series. (there may be more than one answer?)

?, 2 ,3 , 8, 30, 144, 840, 5760, 45360...

(In reply to Puzzle Solution by K Sengupta)

Let us denote the pth term of the sequence by S(p).

In the given sequence, we observe that:

2 divides the 2nd term(2), 3 divides the 3rd term(3), 4 divides the 4th term(8), ........., 9 divides the 9th term (45360)

Accordingly, we can now assert that p divides the pth term, so that:

A(p) = S(p)/p (say)

Then, we must have:

A(2) = 1 = 0!
A(3) = 1 = 1!
A(4) = 2 = 2!
A(5) = 6 = 3!
A(6) = 24 = 4!
A(7) = 120 = 5!
A(8) = 720 = 6!
A(9) = 5040 = 7!

Therefore, A(p) = (p-2)!, so that:
S(p) = (p-2)!*p = p!/(p-1)

Substituting p=1, we have:

S(1) = 1!/0 = 1/0, which is undefined.

Edited on April 29, 2008, 11:33 am

Posted by K Sengupta on 2008-04-29 04:53:58