Heuristic Math Problem Solving Skill: Seeing Pattern

Brilliant tudents know how and when to implement heuristic strategies. Heuristic strategies are devices which drastically limits the search for a solution in a large problem space (Polya, 2004).

They provide a general suggestion or technique for solving different types of problems and can be used independently or in combination. The ten most common strategies include:

Working backwards, Finding a pattern, Adopting a different point of view, Solving a simpler analogous problem, Considering extreme cases, Making a drawing, Intelligent guessing and testing, Accounting for all possibilities, Organizing data and logical reasoning (Posamentier, Smith and Stepelman, 2006).

One must realize that using any of the heuristic skills above mentioned doesn't always guarantee you would seek what you wanted. But, you will be able to gain new insights in your working and that almost always lead to the more surest path to see the light at the end of the tunnel.

Teaching the younger generation and equipping them with heuristic problem solving skills can largely increase their self efficiency and therefore their self confident and self esteem and all these contribute to their ability to face unfamiliar and difficult challenges with cautious, calm, persistent but not fearful nor easily give up.

In this blog post, I will discuss how seeing pattern is considered to be one of the vital heuristic math problem solving skill.
IT is a heuristic practice that can help us make smart choices quickly, find solution wisely and accurately.

According to Alan Schoenfeld:

Mathematics is conceptualized as the science of patterns, an almost empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence.

I will lead the talk by borrowing one good problem, after all, when we want to clarify or explain, it is often best by the use of examples or comparisons.

Problem:

Try to work out the sum of the first few terms from the sequence below:

$1+2(1-x)+3(1-x)(1-2x)+4(1-x)(1-2x)(1-3x)+\cdots+n(1-x)(1-2x)\cdots(1-(n-1)x)$

Now, if the subsequent query is to ask you to work out the formula for the sum of the first $n$ terms for that series, what would you do?

I have hope that you would begin to work this problem out, my whole point is, if you try to attempt at a solution for this particular hard and intriguing math challenge, but to no avail, and when I release of the solution later, you would really learn something from my post and you would realize what costly mistakes you had made and learn to avoid those silly mistakes in the future. Failure in the past while attempting at any hard math challenges is a powerful learning mechanism that yields more robust mathematics learning.

I am looking forward to hearing from you (you are encouraged to comment in this blog post's comment section).