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Work with initial and special cases using barcode development for none control to generate, create none image in none applications.c# upc barcode Many problems will ha none none ve some form of index. For example in problem (iii) the index is n. It can be any natural number and we have to nd integers x, y and z for every possible n.

In these indexed problems you should try to solve the problem in the initial cases, e.g. n = 1, 2 and 3.

This will not necessarily give you the general answer but allows insight and a feel for the problem. In problem (iii) we see that if n = 1 we have to nd solutions to x 2 + y 2 = z. This is easy, take any integers x and y, then x 2 + y 2 is an integer; let s take it to be z.

For example, let x = 3 and y = 5, then x 2 + y 2 = 32 + 52 = 9 + 25 = 34. Thus (x, y, z) = (3, 5, 34) is a possible solution. Recall that the question said show that solutions exist for all n; we don t have to nd all possible solutions for a particular n.

For n = 2 we have to solve x 2 + y 2 = z2 . This is the famous Pythagoras equation, for which we have many solutions, e.g.

32 + 42 = 52 . For n = 3 we have to solve x 2 + y 2 = z3 . Play with this equation and see if you can use the n = 2 case to provide a solution.

I am not suggesting that this will give you an answer; I am suggesting it because it is what I would try, i.e. use the answer in one case to nd the answer in another.

It may or may not work; the point is to try. Can you think of a different method of attack . iPhone OS Work with a concrete case In a similar way to t none for none he above idea, for an abstract problem look at a concrete case. In a problem concerned with sets take a speci c set. So for problem (ii) take a set where X = {a, b, c, d} and Y = {b, d, e, f, g}.

In this case we see that . X. = 4, . Y . = 5. We also see that X Y = . {a, b, c, d, e, f, g}. = 7 and X Y = . {b, d}. = 2. So we need a fo none for none rmula that relates 4, 5, 2 and 7. Play around with some other examples if you don t see a pattern.

Vary X and Y by an element or two and see how the cardinalities change. Be aware though that when proving a general theorem it is not enough to show that it holds in one or two speci c cases. More will be said on this in later chapters.

The examination of speci c cases is intended to provide insight and frequently unlocks the problem.. Draw a picture The human mind is ver none for none y good at working with images. Pictures are excellent for developing intuition about a problem and subsequently suggesting a solution to it. In fact, a diagram is often essential in problems from geometry or physics.

In the case of problem (ii) drawing a Venn diagram is very illuminating.. Devising a plan Think about a similar problem As stated earlier, th e way to become good at solving problems is to solve problems. Similar problems often have similar solutions, so consider problems with similar assumptions or conclusions that you have solved before and see if the same method will work. In the n = 2 case of problem (iii) there is a way of constructing an in nite number of solutions; these solutions are called Pythagorean triples.

It may be worth investigating these to see if the method of nding Pythagorean triples can be adapted to the solution of our problem. (Again, I am not saying it can be, I am saying it is worth thinking about!).
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