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:
= 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.