1.8 MTH 288

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement n2 + 1 ≥ 2n where n is an integer in [1, 4]. Identify the n values for which the equation is to be verified in order to prove the given statement. (You must provide an answer before moving to the next part.)

1, 2, 3, 4

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. The quadratic mean of two real numbers x and y equals (x2+y2)/2−−−−−−−−−√(�2+�2)/2 . By computing the arithmetic and quadratic means of different pairs of positive real numbers, formulate a conjecture about their relative sizes. Rearrange the proof of the statement that the conjecture x2+y22−−−−−√≥x+y2�2+�22≥�+�2 is equivalent to (x - y)2 ≥ 0 in the correct order.

7. This is obtained by applying algebra. 6. (x - y)2 ≥ 0 2. By squaring both sides of the inequality, which is an equivalence transformation on positive numbers, we get 5. This is obtained by rearranging the terms. 1. We conjecture that the quadratic mean is always greater than or equal to the arithmetic mean. Thus, the conjecture is x2+y22−−−−−√≥x+y2�2+�22≥�+�2 . 4. x2 - 2xy + y2 ≥ 0 3. 2x2 +2y2 ≥ x2 + 2xy + y2

Click and drag the statements to prove that between every two rational numbers there is an irrational number.

By finding a common denominator, we can assume that the given rational numbers are a/b and c/b, where b is a positive integer and a and c are integers with a < c. in particular, (a + 1)/b <_ c/b thus, x = (a + (1/2) squareroot 2)/b is between the two given rational numbers, because 0 < squareroot 2 < 2 futhermore, x is irrational, because if x were rational, then 2(bx - a) = square root 2 would be rational this is a contradiction. hence, between every two rational numbers there is an irrational numebr

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that the product of two of the numbers 651000 − 82001 + 3177, 791212 − 92399 + 22001, and 244493 − 58192 + 71777 is nonnegative (for the purpose of this agreement, we will think of 0 as carrying a positive sign). Identify the correct proof of the given statement. (You must provide an answer before moving to the next part.)

Of these three numbers, at least two must have the same sign, since there are only two signs. The product of two with the same sign is nonnegative.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Click and drag the steps to prove min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Assume a is the smallest real number. (Note: In your proof, consider the left side of the equation first.) (You must provide an answer before moving to the next part.)

Suppose a is the smallest of the three real numbers. It follows that a (<_) min(b,c) so, the left-hand side equals a on the right-hand side, min(a, b), c) = min (a, c) on the right-hand side, min(a ,c) = a

Identify the correct proof and the type of proof of the statement that there is a positive integer that equals the sum of the positive integers not exceeding it.

The number 1 has this property, since the only positive integer not exceeding 1 is 1 itself, and therefore the sum is 1. This is a constructive proof, because an example for which the statement is true is given.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that the product of two of the numbers 651000 − 82001 + 3177, 791212 − 92399 + 22001, and 244493 − 58192 + 71777 is nonnegative (for the purpose of this agreement, we will think of 0 as carrying a positive sign). Identify whether the proof is constructive or nonconstructive and the reason for that.

This is a nonconstructive proof, since which product is nonnegative is not identified.

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Identify the set of cases that are required to prove the given statement using proof by cases. (You must provide an answer before moving to the next part.)

a ≤ b ≤ c, a ≤ c ≤ b, b ≤ a ≤ c, b ≤ c ≤ a, c ≤ a ≤ b, c ≤ b ≤ a

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Click and drag the steps to prove min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Assume b is the smallest real number. (Note: In your proof, consider the left side of the equation first.) (You must provide an answer before moving to the next part.)

suppose b is the smallest of the three real numbers it follows that b <_ min(b, c). thus, min(a, b) = b so, the left-hand side equals b on the right-hand side, min (a, b) = b, so, min(min(a, b), c) = min(b, c) on the right-hand side, min(b, c) = b

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement that min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Click and drag the steps to prove min(a, min(b, c)) = min(min(a, b), c) whenever a, b, and c are real numbers. Assume, c is the smallest real number. (Note: In your proof, consider the left side of the equation first.)

suppose c is the smallest of the three real numbers it follows that c <_ min(b, c) so, the left-hand side equals c on the right-hand side, c <_ min(a, b), so min(min(a, b), c) = c therefore, on the right-hand side equals c

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. Consider the statement n2 + 1 ≥ 2n where n is an integer in [1, 4]. The correct proof of the given statement is "For n = 1, 12 + 1 = 2 ≥ 2 = 21; for n = 2, 22 + 1 = 5 ≥ 4 = 22; for n = 3, 32 + 1 = 10 ≥ 8 = 23; and for n = 4, 42 + 1 = 17 ≥ 16 = 24." (You must provide an answer before moving to the next part.)

true

NOTE: This is a multi-part question. Once an answer is submitted, you will be unable to return to this part. The quadratic mean of two real numbers x and y equals (x2+y2)/2−−−−−−−−−√(�2+�2)/2 . By computing the arithmetic and quadratic means of different pairs of positive real numbers, formulate a conjecture about their relative sizes. Formulate the conjecture about the relative sizes of x and y and the evidences examined for formulating the conjecture. (You must provide an answer before moving to the next part.)

x2+y22−−−−−√≥x+y2�2+�22≥�+�2 If x = 1 and y = 10, then their arithmetic mean is 5.5 and their quadratic mean is 7.11, and if x = 5 and y = 8, then their arithmetic mean is 6.5 and their quadratic mean is 6.67.