answer:1. Identify the radii of the two circles and the given distance between the points on the circles. - Radius of the larger circle ( R = 28 ) - Radius of the smaller circle ( r = 12 ) - Distance between the points on the circles ( d = 10 ) 2. Let ( O ) be the center of the larger circle and ( O_1 ) be the center of the smaller circle. 3. Considering the nature of the problem, we are given that one circle is inside the other, and we need to determine the distance between their centers ( OO_1 ). 4. Place the center of the larger circle ( O ) at the origin and place the center of the smaller circle ( O_1 ) at a point on the x-axis to simplify the geometry. 5. Draw the diameter of the larger circle so that it passes through the center of the smaller circle. This is possible as the smaller circle is inside the larger circle. 6. Let the points where the diameter intersects the circumference of the larger circle be ( A ) and ( B ). Let the points where the diameter intersects the circumference of the smaller circle be ( C ) and ( D ). 7. The sum of the radii is: [ R + r = 28 + 12 = 40 ] 8. The difference between the radii is: [ R - r = 28 - 12 = 16 ] 9. The sum of the distances AC and CB is equal to the distance AB, which covers the entire diameter of the larger circle: [ AB = 2R = 2 times 28 = 56 ] 10. The distance ( CD ) is given as ( d ): [ CD = 10 ] 11. The distances from the center ( O ) to the points ( A ) and ( C ) are: - ( OA = R = 28 ) - ( OC = OA - d = 28 - 10 = 18 ) 12. The distance ( O_1D ) from ( O_1 ) to ( D ) where ( D ) is the second point of tangency on the smaller circle: - [ O_1D = r = 12 ] 13. The entire distance from ( O ) to ( O_1 ) involves summing the distance ( OC ) and the inner radius ( r ): - [ OC + O_1D = 18 + 12 = 30 ] 14. However, due to the relative positions and distance algebraically working it out, calculating the entire distance would be less since calculations show: - [ OO_1 = sqrt{30^2 - 28^2} = sqrt{36} = 6 ] 15. We conclude the intercessory distance is algebraically calculated twice presenting: - [ OO_1 = 6 ] # Conclusion [ boxed{6} ]

answer:(a) **Prove the recurrence relation for Fibonacci numbers** To prove the recurrence relation ( F_n = F_{n-1} + F_{n-2} ) for ( n geq 2 ), consider the number of ways to represent ( n ) as an ordered sum of 1's and 2's. 1. **Step 1**: Note that each representation starts either with a 1 or a 2. 2. **Step 2**: If the first term is 1, the remaining ( n-1 ) terms must sum to ( n-1 ), and there are ( F_{n-1} ) ways to do this. 3. **Step 3**: If the first term is 2, the remaining ( n-2 ) terms must sum to ( n-2 ), and there are ( F_{n-2} ) ways to do this. 4. **Step 4**: Combining these two possibilities, we get ( F_n = F_{n-1} + F_{n-2} ). Hence, the recurrence relation is proved. (blacksquare) (b) **Derive the closed-form expression for Fibonacci numbers** To derive the closed-form expression, also known as Binet's formula, we start with the characteristic equation of the recurrence relation ( F_n = F_{n-1} + F_{n-2} ). 1. **Step 1**: Assume a solution of the form ( F_n = alpha^n ). 2. **Step 2**: Substitute ( F_n = alpha^n ) into the recurrence relation: [ alpha^n = alpha^{n-1} + alpha^{n-2} ] Dividing through by ( alpha^{n-2} ): [ alpha^2 = alpha + 1 ] 3. **Step 3**: Solve the quadratic equation: [ alpha^2 - alpha - 1 = 0 ] Using the quadratic formula ( alpha = frac{b pm sqrt{b^2 - 4ac}}{2a} ): [ alpha_{1,2} = frac{1 pm sqrt{5}}{2} ] Let ( alpha_1 = frac{1 + sqrt{5}}{2} ) and ( alpha_2 = frac{1 - sqrt{5}}{2} ). 4. **Step 4**: General solution for ( F_n ) is a linear combination of the solutions: [ F_n = A alpha_1^n + B alpha_2^n ] Use initial conditions ( F_0 = 1 ) and ( F_1 = 1 ): [ F_0 = 1 = A cdot 1 + B cdot 1 implies A + B = 1 ] [ F_1 = 1 = A alpha_1 + B alpha_2 implies A frac{1 + sqrt{5}}{2} + B frac{1 - sqrt{5}}{2} = 1 ] 5. **Step 5**: Solve the system of equations: [ A + B = 1 ] [ A frac{1 + sqrt{5}}{2} + B frac{1 - sqrt{5}}{2} = 1 ] Subtract the first equation multiplied by (frac{1 + sqrt{5}}{2}): [ A cdot sqrt{5} + B cdot (-sqrt{5}) = sqrt{5} ] [ A - B = frac{1}{sqrt{5}} ] Solve the linear system: [ A + B = 1 ] [ A - B = frac{1}{sqrt{5}} ] Adding these equations: [ 2A = 1 + frac{1}{sqrt{5}} = frac{sqrt{5} + 1}{sqrt{5}} ] [ A = frac{sqrt{5} + 1}{2sqrt{5}} = frac{1 + sqrt{5}}{2 sqrt{5}} ] Subtracting these equations: [ 2B = 1 - frac{1}{sqrt{5}} = frac{sqrt{5} - 1}{sqrt{5}} ] [ B = frac{sqrt{5} - 1}{2sqrt{5}} = frac{1 - sqrt{5}}{2 sqrt{5}} ] 6. **Step 6**: Substitute ( A ) and ( B ): [ F_n = frac{1}{sqrt{5}} left[ left( frac{1 + sqrt{5}}{2} right)^n - left( frac{1-sqrt{5}}{2} right)^n right] ] But we need ( F_n ): [ F_n = frac{1}{sqrt{5}} left[ left( frac{1 + sqrt{5}}{2} right)^{n+1} - left( frac{1 - sqrt{5}}{2} right)^{n+1} right] ] Thus, the closed-form formula is: [ F_n = frac{1}{sqrt{5}} left[ left( frac{1 + sqrt{5}}{2} right)^{n+1} - left( frac{1 - sqrt{5}}{2} right)^{n+1} right] ] (blacksquare)

answer:Let's analyze each option step by step to determine which condition cannot determine that triangle ABC is a right triangle. **Option A:** Given angle A + angle B = angle C, and knowing that in any triangle, the sum of angles is 180^{circ}, we have: [ angle A + angle B + angle C = 180^{circ} ] Substituting angle A + angle B with angle C, we get: [ angle C + angle C = 180^{circ} implies 2angle C = 180^{circ} implies angle C = 90^{circ} ] This means triangle ABC must be a right triangle with angle C as the right angle. Therefore, option A does not meet the requirement of not being able to determine if triangle ABC is a right triangle. **Option B:** Given angle A : angle B : angle C = 1:2:3, and knowing the sum of angles in a triangle is 180^{circ}, we can express this as: [ angle A + angle B + angle C = 180^{circ} ] Given the ratio, angle C can be expressed as 180^{circ} times frac{3}{6} = 90^{circ}, since 1+2+3=6 and angle C is frac{3}{6} of the total. This means triangle ABC must be a right triangle with angle C as the right angle. Therefore, option B does not meet the requirement. **Option C:** Given a^{2} = b^{2} - c^{2}, we can rearrange this to fit Pythagoras' theorem: [ a^{2} + c^{2} = b^{2} ] This is the Pythagorean theorem, which directly implies that triangle ABC is a right triangle with the right angle opposite side b. Therefore, option C does not meet the requirement. **Option D:** Given a^{2} = 5, b^{2} = 12, and c^{2} = 13, we check if these satisfy the Pythagorean theorem: [ a^{2} + b^{2} = 5 + 12 = 17 neq c^{2} ] Since c^{2} = 13, the sum of a^{2} and b^{2} does not equal c^{2}, which means triangle ABC cannot be determined to be a right triangle using these side lengths. Therefore, option D meets the requirement of not being able to determine if triangle ABC is a right triangle. Hence, the correct answer is boxed{D}.

answer:Since f(x) = (m^2 - 2m + 1)x^{2m-1} is a power function, we have m^2 - 2m + 1 = 1. Solving this equation yields m = 0 or m = 2. Let's analyze each case: 1. When m = 0, the function becomes f(x) = x^{-1}. On the interval (0, +infty), this function is a decreasing function, which does not meet the requirement of being an increasing function. 2. When m = 2, the function becomes f(x) = x^{3}. On the interval (0, +infty), this function is an increasing function, which satisfies the given condition. Therefore, the answer is boxed{2}. To solve this problem, we first set up an equation using the given power function to find the possible values of m. Then, we verify each value to see if it meets the condition of being an increasing function on the specified interval. This problem tests the understanding of power functions and their applications, making it a fundamental question.