answer:1. We start by recognizing that we need to show that for two adjacent rational numbers frac{a}{b} and frac{c}{d} in the Farey sequence F_n (where a/b < c/d), the condition |bc - ad| = 1 holds. 2. First, recall that the Farey sequence F_n consists of all fractions in lowest terms between 0 and 1 with denominators less than or equal to n, arranged in increasing order. 3. We aim to prove the statement by induction on n. 4. **Base Case (n = 3)**: - For n = 3, the Farey sequence F_3 is: left{ 0, frac{1}{3}, frac{1}{2}, frac{2}{3}, 1 right} - Checking each pair of adjacent fractions: - For frac{1}{3} and frac{1}{2}: [ left| 1 cdot 2 - 1 cdot 3 right| = left| 2 - 3 right| = 1 ] - For frac{1}{2} and frac{2}{3}: [ left| 1 cdot 3 - 2 cdot 2 right| = left| 3 - 4 right| = 1 ] - Hence, the base case holds as |bc - ad| = 1 is satisfied. 5. **Inductive Step**: - Suppose the statement is true for F_{n-1}. We need to show it remains true for F_n. - When increasing n to n+1, new fractions of the form frac{k}{n+1} are added to F_n. - New fractions cannot exist between any two fractions where |bc - ad| > 1. Hence, they only appear where |bc - ad| = 1 by the induction hypothesis. 6. Consider adjacent terms frac{a}{b} and frac{c}{d} in F_{n+1} not both inherited from F_n. - Assume frac{a}{b} < frac{k}{n+1} < frac{c}{d}; - We need to show A=k b - a (n+1) and B=c (n+1) - d k are both equal to 1: [ A = k b - a (n+1) ] [ B = c (n+1) - d k ] 7. By the inductive hypothesis, |b c - a d| = 1. Suppose one of A or B is greater than 1: [ b + d < b B + d A = (b c - a d) (n+1) = (1)(n+1) = n+1. ] - This implies b + d < n + 1, consistent with the Farey sequence construction rules, ensuring new terms retain the property |bc - ad| = 1, proving by contradiction. 8. Therefore, assuming the original assumptions hold, the implication extends perfectly, ensuring frac{a}{b} and frac{c}{d} remain adjacent with |bc - ad| = 1 throughout. # Conclusion: [ boxed{|bc - ad| = 1} ]

answer:To find the perimeter of a semicircle, you need to add the length of the straight edge (the diameter) to the length of the curved edge (half the circumference of a full circle). The diameter of the semicircle is twice the radius, so: Diameter = 2 * Radius = 2 * 7 = 14 units The circumference of a full circle is given by the formula: Circumference = 2 * π * Radius For a semicircle, we only want half of this circumference, so: Half Circumference = π * Radius = π * 7 Now, add the diameter and half the circumference to get the perimeter of the semicircle: Perimeter = Diameter + Half Circumference Perimeter = 14 + (π * 7) Using the approximate value of π as 3.14159, we get: Perimeter ≈ 14 + (3.14159 * 7) Perimeter ≈ 14 + 21.99113 Perimeter ≈ 35.99113 units So, the approximate perimeter of the semicircle is boxed{35.99} units.

answer:1. Given the function ( f ) satisfies ( f(cos x) = cos (17x) ) for every real ( x ). 2. We need to show that ( f(sin x) = sin (17x) ) for every ( x in mathbb{R} ). To prove this, we can use the trigonometric identity that relates sine and cosine: [ sin x = cos left( frac{pi}{2} - x right) ] 3. Substitute ( sin x ) with ( cos left( frac{pi}{2} - x right) ) in the function ( f ): [ f(sin x) = fleft( cos left( frac{pi}{2} - x right) right) ] 4. Using the given property of the function ( f ): [ fleft( cos left( frac{pi}{2} - x right) right) = cos left( 17 left( frac{pi}{2} - x right) right) ] 5. Simplify the argument of the cosine function: [ cos left( 17 left( frac{pi}{2} - x right) right) = cos left( frac{17pi}{2} - 17x right) ] 6. Use the trigonometric identity ( cos left( frac{pi}{2} - theta right) = sin theta ): [ cos left( frac{17pi}{2} - 17x right) = sin (17x) ] Therefore, we have: [ f(sin x) = sin (17x) ] (blacksquare)

answer:To find the number, we can set up the equation as follows: 144 / 0.144 = x / 0.0144 Now, we can cross-multiply to solve for x: 144 * 0.0144 = x * 0.144 Now, we calculate the left side of the equation: 144 * 0.0144 = 2.0736 Now we have: 2.0736 = x * 0.144 To solve for x, we divide both sides of the equation by 0.144: x = 2.0736 / 0.144 x = 14.4 Therefore, the number is boxed{14.4} .