Prove that xn yn has x y as a factor for all positive integers n. in VS .NET

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1.30. Prove that xn yn has x y as a factor for all positive integers n.
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The statement is true for n 1 since x1 y1 x y. Assume the statement true for n k, i.e., assume that xk yk has x y as a factor. x
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x x y y xk yk The rst term on the right has x y as a factor, and the second term on the right also has x y as a factor because of the above assumption. Thus xk 1 yk 1 has x y as a factor if xk yk does. Then since x1 y1 has x y as factor, it follows that x2 y2 has x y as a factor, x3 y3 has x y as a factor, etc.
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1.31. Prove Bernoulli s inequality 1 x n > 1 nx for n 2; 3; . . . if x > 1, x 6 0.
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The statement is true for n 2 since 1 x 2 1 2x x2 > 1 2x. Assume the statement true for n k, i.e., 1 x k > 1 kx. Multiply both sides by 1 x (which is positive since x > 1). Then we have 1 x k 1 > 1 x 1 kx 1 k 1 x kx2 > 1 k 1 x Thus the statement is true for n k 1 if it is true for n k. But since the statement is true for n 2, it must be true for n 2 1 3; . . . and is thus true for all integers greater than or equal to 2. Note that the result is not true for n 1. However, the modi ed result 1 x n A 1 nx is true for n 1; 2; 3; . . . .
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MISCELLANEOUS PROBLEMS 1.32. Prove that every positive integer P can be expressed uniquely in the form P a0 2n a1 2n 1 a2 2n 2 an where the a s are 0 s or 1 s.
Dividing P by 2, we have P=2 a0 2n 1 a1 2n 2 an 1 an =2. Then an is the remainder, 0 or 1, obtained when P is divided by 2 and is unique. Let P1 be the integer part of P=2. Then P1 a0 2n 1 a1 2n 2 an 1 . Dividing P1 by 2 we see that an 1 is the remainder, 0 or 1, obtained when P1 is divided by 2 and is unique. By continuing in this manner, all the a s can be determined as 0 s or 1 s and are unique.
1.33. Express the number 23 in the form of Problem 1.32.
The determination of the coe cients can be arranged as follows: 2 23 2 11 2 5 2 2 2 1 0
Remainder Remainder Remainder Remainder
1 1 1 0
Remainder 1
The coe cients are 1 0 1 1 1. Check: 23 1 24 0 23 1 22 1 2 1. The number 10111 is said to represent 23 in the scale of two or binary scale.
1.34. Dedekind de ned a cut, section, or partition in the rational number system as a separation of all rational numbers into two classes or sets called L (the left-hand class) and R (the right-hand class) having the following properties: I. II. III. The classes are non-empty (i.e. at least one number belongs to each class). Every rational number is in one class or the other. Every number in L is less than every number in R.
Prove each of the following statements: (a) There cannot be a largest number in L and a smallest number in R. (b) It is possible for L to have a largest number and for R to have no smallest number. type of number does the cut de ne in this case What
(c) It is possible for L to have no largest number and for R to have a smallest number. What type of number does the cut de ne in this case
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