1. B Prove
that
(n+y+z)² = (y + 2 - 2)² + (2+2-4)3 + (a+y-2) 3+ 29xyz
Answers
Answer:
If xy +xz + yz = x^2 + y^2 + z^2, what is [(x -y) ^2+ (x+y+z) ^2] /9xyz?
The only way the first equation can hold is if x = y = z which is intuitively hit upon. The proof is far more laborious with multiple unknown variables and can constitute a short book, and well beyond the scope of this short article. In this situation x, y, and z, can be any real number or imaginary number, except 0, but at all times equal to each other. Say it is 1, then in the first equation the left side will be 1 + 1 + 1 equal to the right side which is 1 + 1+ 1 also, hence it is 3 = 3. So specifically if x = y = z = 1, then the next equation will look as follows : 0 + 3^2/9•1•1•1 which is 9/9 which is 1. Now if x = y = z = 0, then the denominator becomes zero and it becomes undefined, therefore neither x, nor y, nor z can be zero effectively speaking. If the number is 2, then the first equation becomes 12 = 12, and the second equation becomes 6^2/9•8 which is 36/72 which is 1/2. (Please note that x-y squared is zero in each case, in the second equation). Let’s see what happens when the number is 3. Then the first equation will give you 27 = 27, and the second equation will give you the following : 9^2/9•3•3•3 which is 81/243 which is 0.333…. I am going to restrict the intuitive study here to positive and negative integers (whole numbers) only, excluding fractions to see what happens with this set of numbers. As the positive integer increases the first equation will give larger and larger number, and second equation will give smaller and smaller number approaching zero. All values will fall between 1 and 0 for the positive real numbers with x, y, and z being equal to each other. Greater the x, y, or z (all being equal), smaller will be the solution to the second equation. Now I have talked about positive whole numbers (integers) here so far. What happens if I use negative whole numbers, then the calculated values for the first equation does not change as the squared items turn the negative base numbers into positive numbers, but the calculated values for the second equation will give you a negative number as an answer between -1 and 0. This is because in the second equation the denominator has the triple product x•y•z which will give you a negative number (-1 • -1 • -1 is -1 as an example). If you have noticed the first equation gives a divergent series with both positive and negative integers, but the second equation gives a convergent series between 1 and 0 with positive integers, and between - 1 and 0 for negative integers. It is because of the mathematical structures of the two equations, capable of giving divergent, and convergent series respectively. And what about if x = y = z = i which is square root of - 1 (imaginary number) then the first equation will see real numbers at the end of the operation, as i • i or i^2 is - 1, but the second equation will have real number in the numerator, and complex number in the denominator, since the x•y•z triple product in the denominator will give you - i times the 9 which is -9 i. So if x = y = z = i (square root of - 1), then i • i is - 1 and - 1 • i is - i, which will give you - 9 i in the denominator, and - 9 in the numerator, giving you - 9/- 9 i which is 1/i which is a complex number (it is also inverse of i). The above will be extremely hard to mathematically prove (with multiple variables in place) that x = y = z must hold here for the first equation to exist as a valid mathematical relationship in the first place, so seeing through number examples and using mathematical intuition, we can make sense of the first equation well, and what will be the correlates for the second equation and the possible results. One can plot a graph of the results of the positive and negative whole numbers for the first equation and the results of the second equation which will show that as the positive whole number (integer) increases from 1 to positive infinity, the second equation will yield results that will approach 0 starting from 1, and as the negative whole numbers decreases from -1 to negative infinity, the second equation will give results from - 1 to values approaching zero ! This is a great example of one equation giving you a divergent series, and the second equation contingent on the first yielding an asymptotic convergent series…(Using fractions for the x, y, and z will give you completely different results and the second equation can give you larger and larger positive values for smaller and smaller positive fractions and the converse, and same second equation will give you smaller and smaller negative values, for larger and larger and larger negative fractions, and the converse without a restricted narrow band of 1 to 0 , or 0 to - 1 unlike the whole numbers !). Kaiser T, MD (Life long physics, math, cosmology, and science proponent).