What is the next number in this sequence?
1, 4, 9, 16, 25, pi
The sequence is not arbitrary. Create a mathematical function that generates this sequence.
(In reply to The 5th degree vs. The interrupted sequence
by Brian Smith)
It is readily observed that all the other terms except the 6pth term for p=1,2,.... is a perfect square.
Thus, the 6pth term is a real number which is not a perfect square.
Accordingly, if S(x) denotes the xth term of the given sequence, then:
S(x) = x^2, whenever x!= 0 Mod 6, and:
S(x) = r_i, whenever x = 0 Mod 6, and:
r_i is a real number which is not a perfect square whenever i = 6, 12, 18, .....
Therefore, S(7) = 7^2 = 49, and the given sequence could continue as:
r_1 = pi, and r_j for all j> 1 is a real number which is not a perfect square.
Note: The above exercise is merely an elaboration upon part II of the excellent methodology submitted by Brian Smith.