In a spirit of developing scientific art and brings intelligence and exercise.
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This is an attempt to show how mathematical multiplication and division sometimes can be viewed geometrically without getting lost, and solving something at the end of the day. Armed with patience and pencil, one may actually sometimes get a chance to see math inner algebraic workings "in action", and solve particular problem at the end of the day.
Sketch shows an alternative to mathematical induction method in cases of solving square/cubic expression equalness in math. Exercise was to find whether for all natural number sequence of "n" (where user chooses on which natural number n ends) an expression of 1^3 + 2^3 + 3^3 + ... + n^3 is equal and the same as expression ( n^2 * (n + 1)^2 ) / 4.
This is exercise which i took from old soviet mathematics book for schools on my own initiative, only to discover that with patience and some trickery you can crack some tasks and enjoy the process. It is crime what modern schools do with their teaching methods to resulting perception of math in heads of those, to whom it is taught, planting the seeds of hate towards mathematics, which in turn plants next deeper seeds for uninformed, lazy and shallow society, with all the consequential dangers to itself...
What is going on there is this - we "draw" sum of cubic elements on left side of problematic equation for some fairly low n value, say 5. We draw cube of "generic size" 1 + cube of generic size 2 + cube of generic size 3 and so on. It is then, when one can realize that task asks us whether a summary volume of such assembly is the same as volume of assembly made differently. Then we start operating on right sight of problematic equation for the same n = 5 value. We first "engineer" the multiplication in upper part into an assembly of square according to definition of what it means to multiply some area on given distance - a sort of spatial extruding operation, which gives us volume. We then divide by 4, or into 4 equal pieces, the resulting geometric construction, keeping in mind what it's resulting volume in generic units of measure was due to prior operation. The number of generic volume units on left side of problematic equation has to be numerically equal to number of generic volume units on right side.
Such approach enables one to "see" solution and how it takes shape during process to some degree. In this approach during exercise whole procedure utilizes your brain GPU - "graphics" processing unit/ and neurons meant for visual abstraction exercise and cognition - which otherwise get lend for attention to only funny cats videos, advertising, naked advertising or other detrimental commercials at best. If you train patiently like this, no matter how hard it is and how boring schools made math look like to you - it is then your cognition and mind processes becomes ,thanks to conscious efforts, more of your custom-made property instead of commercially owned and hijacked brain.
Do the math at your own pace and speed, take easy or difficult tasks - whichever even if you are beyond 60. Find old school books and get in a habit of training on whatever school/college/university grade level. Awake your scientific, technical curiosity..
Resist, and reclaim your brain...
(12 June 2018)