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 Who Ordered the Pi? (Posted on 2003-04-21)
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.

 See The Solution Submitted by Bryan Rating: 1.7778 (9 votes)

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 Alternative Methodology Comment 11 of 11 |
(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:

1,4,9,16,25,r_1,49,64,81,100,121,r_2,169,196,225,256,289,r_3,
....    with:
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.

 Posted by K Sengupta on 2008-04-02 05:54:31
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