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Who Ordered the Pi? (Posted on 2003-04-21) Difficulty: 4 of 5
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)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts re: The 5th degree vs. The interrupted sequence | Comment 10 of 11 |
(In reply to The 5th degree vs. The interrupted sequence by Brian Smith)

True. Part I of your methodology gives the precise formula for the given sequence.

Also, in my opinion, your  solution is correct, since the official solution is inclusive of some inadvertent typographical anomalies.

The appropriate methodology in support of this assertion  is furnished hereunder as follows:

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

Then, we observe that:
S(1) = 1 = 1^2
S(2) = 4 = 2^2
S(3) = 9 = 3^2
S(4) = 16 = 4^2
S(5) = 25 = 5^2
S(6) = pi = 6^2 + (pi-36)
               
This leads us to assert that S(p) = p^2 + (pi-36)*G(p) ....(*)

In (*), we observe that G(p) = 0, whenever  p = 1,2,3,4,5 and:
G(p) = 1, if G(p) = 6

This is satisfied for G(p) = (p-1)(p-2)(p-3)(p-4)(p-5)/5!

Now, G(7) = 6!/5! = 6, so that:

S(7) = 7^2 + (pi-36)*6
= 49 + 6*pi - 216
= 6*pi - 167

Thus, it is now apparent, that in  the final steps of the official solution, for n=7,it should have been  "49 + (pi-36)6!/5!" rather than "36 + (pi-36)6!/5!" leading to "6*pi - 167" instead of "6*pi - 180".

Edited on April 2, 2008, 6:02 am
  Posted by K Sengupta on 2008-03-31 14:58:22

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