All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars
 perplexus dot info

 Polynomial of degree 3n (Posted on 2016-05-07)
A polynomial P(x) of degree 3n has the value 2 at x= 0, 3, 6,..., 3n, and:
the value 1 at x= 1, 4, 7, ... , 3n-2, and:
the value 0 at x= 2, 5, 8, ... , 3n-1.

Given that P(3n+1) = 730, find n.

 No Solution Yet Submitted by K Sengupta Rating: 4.0000 (2 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
 Possible Solution | Comment 5 of 6 |
n = 4

...2  1   0   2   1   0   2   1   0   2  1  0  2  730

..-1  -1  2  -1  -1   2  -1   -1  2  -1 -1  2  728
...0  3  -3  0   3   -3   0   3  -3  0  3  726
...3 -6  3   3  -6    3   3   -6  3  3  723
...-9  9  0  -9    9   0   -9  9  0  720
...18 -9  -9  18  -9  -9   18 -9  720
...-27  0  27 -27  0  27  -27  729
...27 27 -54  27  27 -54  756
...0  -81  81  0  -81  810
...-81 162 -81 -81  891
...243 -243  0  972
...-486  243 972
...729  729
...0

The top row shows the given values of P(x), at unit intervals, the
last value being P(3n+1). The rows below show the first differences
to the 13th differences. Dots to the left indicate that we donâ€™t yet
know the value of n and therefore how many three-cycles to
include in each set.
However, the zero that appears as a thirteenth difference, is the
first one to be at the tip of a triangle that extends upwards and
is derived from a set of P(x) values including 730.
A 13th difference being zero shows that a polynomial of degree 12
can pass through the 14 points represented on the top row,
so 3n = 12, giving n = 4.

Perhaps there are bigger values of n that will work..?

 Posted by Harry on 2016-05-08 14:58:49
Please log in:
 Login: Password: Remember me: Sign up! | Forgot password

 Search: Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (1)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2018 by Animus Pactum Consulting. All rights reserved. Privacy Information