How are patterns used to create algebraic expressions?

1 Answer
Jun 20, 2015

Answer:

This question is rather general, so I will only address a small fragment of it...

Explanation:

How do you identify patterns and express them as algebraic expressions?

For example, given the sequence #1, 4, 10, 20, 35, 56#

What is the pattern? How do you find the next number in the sequence? What is the formula for the #n^(th)# term in the sequence?

With these sort of problems, it is often helpful to construct a sequence of differences between successive terms, repeating this process to see if you end up with a constant sequence...

#1, 4, 10, 20, 35, 56#
# -> 3, 6, 10, 15, 21#
# -> 3, 4, 5, 6 #
# -> 1, 1, 1 #

Since it has taken #3# steps to get to a constant sequence, the original sequence is expressible as a cubic expression.

We can directly construct the formula for #a_n# from the first term of each of these sequences as:

#a_n = color(red)(1) + color(red)(3) * n/(1!) + color(red)(3) * (n(n-1))/(2!) + color(red)(1) * (n(n-1)(n-2))/(3!)#

I rather like this discrete variant on Taylor's theorem.

This approach will not work well with the Fibonacci sequence, since it is essentially exponential, not polynomial...

#0, 1, 1, 2, 3, 5, 8, 13, 21, 34#
#-> 1, 0, 1, 1, 2, 3, 5, 8, 13#
#-> -1, 1, 0, 1, 1, 2, 3, 5#
#-> 2, -1, 1, 0, 1, 1, 2#
#-> -3, 2, -1, 1, 0, 1#
#-> 5, -3, 2, -1, 1#
...

But it is possible to find a formula for #F_n#, viz

#F_n = (phi^n - (-phi)^-n)/sqrt(5)#

where #phi = (1+sqrt(5))/2#

Even the area of numeric sequences is too big to give a full answer.