There is a sequence defined by using common mathematical operations.
The first two terms in the sequence are 10 and 12 and the sixth term is 60. The seventh term, however, is undefined.
What are the third, fourth and fifth terms?

10, 12, __, __, __, 60, (?)

Halve the values of these terms:
 5, 6, __, __, __, 30, (?)

If the difference between terms is:
1, 3, 5, 7, ....., a sequence of odd numbers,
then the sequence is
 5, 6, 9, 14, 21, 30, 41, 54,...

The summation of the series of odd numbers is given by x^2. 
The value of each term under investigation for this sequence can be given by: 
(5 + x^2) where x >= 0.

The presented sequence however is double this so the expression needs to be:

but the 7th term of this series is 82, and therefore defined.

Division by zero is "undefined".
How to divide the term by zero in terms of the defining x value?
If x=6 then 2(5+x^2) = 82.  If a denominator (6-x) is defined and x=0 then the denominator is zero.

Let each term of the sequence be mutiplied by 1, expressed as (6-x)/(6-x) which is defined for all x except 6.

The expression required then is:
2*(6-x)(5+x^2)/(6-x)  where x>=0 and x=6 is undefined.

The following sequence:
10, 12, 18, 28, 42, 60, (82?), 108, 138, ..........  fulfills the problems requirements.


More generally this sequence can be represented as:
a*(b-x)(c+x^2)/(b-x) where all values are integers.
     a is a multiplier which provides an offset from the base sequence - (c+x^2)
     c is the first term of the base sequence
     x is any integer >= 0
     b is the value for which (b+1) is the term which will be "undefined".

Even more generally 'a' can be any multiplier and 'x^2' can be any defining interval.

The important thing is the (b-x) factor as it provides the indeterminate value, ie, (b-x)/(b-x) for b=x!!

Posted by brianjn on 2008-03-14 21:15:37