What two numbers come next in this sequence, and what is the rule?

1/2, 2, 9, 48, 300, __, __.

(In reply to

Puzzle Solution With Explanation by K Sengupta)

Let us denote the pth term of the sequence as S(p), and define T(p) = S(p+1)/S(p)

Then, from the given values of S(p), we observe that:

T(1) = 4

T(2) = 9/2

T(3) = 16/3

T(4) = 25/4

T(5) = 36/5

Accordingly, it is now apparent that T(p) = (p+1)^2/p, so that:

S(p+1) = [(p+1)^2/p]*S(p)

Let us define U(p) = S(p)/p........(I)

Then, from (I) we must have:

U(p+1)

= (p+1)U(p)

= (p+1)p*U(p-1)

= .......

= (p+1)*p*....*2*U(1)

= c*(p!), where c = U(1), say.

But, then, c = S(1)/1 = S(1) = 1/2(given)

Accordingly, U(p) = (1/2)*(p!), so that:

S(p)/p = (1/2)*(p!),giveng:

S(p) = (p/2)*(p!).......(II)

Substituting p =1,2,....,5 in turn in (II) this is easily verified that this is in conformity with the value of each term as provided in the problem text.

Consequently, substituting p = 6, 7 we obtain the required next two terms as 2160 and 17640.