Given the interest in sequences it seems fitting to run
another series of sequence puzzles of a different (and
more challenging) nature. For each of the sequences that follow you
should find a formula, using the variable
n, that produces the
terms in the sequence starting with n=1. For example, if the
sequence were:
2/3, 3/4, 4/5, 5/6, 6/7, ...
a formula would be (n+1)/(n+2) since for n=1, 2, 3, ... this
formula produces
exactly the terms in the given sequence. The
sequence:
0, 3, 8, 15, 24, 35, 48, 63, 80,...
is produced
by n^2-1.
Here are the sequences for you to find formulas for.
6, 9, 14, 21, 30, 41, 54, 69, 86, ...
-1, 1, 5, 13, 29, 61, 125, 253, 509, ...
1/2, 3/5, 2/3, 5/7, 3/4, 7/9, 4/5, 9/11, ...-1, 8, -27, 64, -125, 216, -343, ...
0, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, ...
Even when a pattern is very clear or simple, a formula that
produces the terms can be very hard to find.
The sequence 1, 2,
3, 5, 8, 13, 21, 34, 55, ... ( the famous Fibonacci
sequence) given last issue has
a very clear pattern. The formula
for these terms, however, is so complicated that finding it
without using some
fairly sophisticated mathematical techniques
is practically impossible (believe it or not, the formula
involves
the square root of 5). As a last and very difficult
puzzle, find a formula for the following sequence (that has a
very obvious pattern):
-1, -1, 1, 1, -1, -1, 1, 1, -1, -1, ...