answer:The infinite product prod_{n=1}^{infty}left(1+frac{1}{2^n}right) converges to approximately 2.384231029. This can be shown using various methods, such as taking the logarithm of the product and using the Taylor expansion of the logarithm function. The product can also be expressed in terms of the Euler's function and the q-Pochhammer symbol.

answer:To calculate the time of flight ( t ) of the rubber band, we can use the kinematic equation under the assumption that the rubber band's final height is the same as its starting height and air resistance is negligible. We consider the upward direction as positive. Given: - Initial velocity ( u = 25 ) m/s - Final vertical displacement ( s = 0 ) m (since it returns to the starting height) - Acceleration due to gravity ( a = -g = -9.8 ) m/s² Using the equation ( s = ut + frac{1}{2}at^2 ), we can solve for ( t ): [ 0 = (25)t + frac{1}{2}(-9.8)t^2 ] Rearrange the equation to solve for ( t ): [ 0.5(-9.8)t^2 + 25t = 0 ] Divide by 0.5(-9.8) to simplify: [ t^2 + frac{50}{9.8}t = 0 ] Simplify further: [ t^2 + frac{50}{9.8}t = 0 ] Factor out ( t ): [ t(t + frac{50}{9.8}) = 0 ] This gives us two possible solutions, ( t = 0 ) (initial launch) and ( t = -frac{50}{9.8} ). Since we are looking for the time it takes to fall back, we take the positive time value: [ t approx -frac{50}{9.8} ] Convert to seconds: [ t approx frac{50}{9.8} approx 5.10 text{ s} ] Therefore, the time of flight of the rubber band is approximately ( 5.10 ) seconds.

answer:We can use the information given in the diagram to solve this problem. Let x, y, and z be the numbers in the shaded circles. Then, we have the following equations: x + y + z = 21 x + 2 + 6 = 8 y + 6 + 4 = 14 z + 4 + 3 = 14 Adding the last three equations, we get: 2x + 2y + 2z = 36 Dividing both sides by 2, we get: x + y + z = 18 Subtracting this equation from the first equation, we get: 0 = 3 This is a contradiction, so there is no solution to this problem.

answer:The characteristic polynomial of the given matrix can be calculated using the determinant of the matrix with the main diagonal elements replaced by x and the other elements replaced by 0. Let's denote the matrix as A and proceed with the calculation. The characteristic polynomial p(x) is given by p(x) = det(A - xI), where I is the identity matrix. After performing the calculations, we get the characteristic polynomial to be: [ p(x) = x^3 - left(frac{49}{5} + frac{11}{2} - frac{29}{10}right)x^2 + left(frac{49}{5} cdot frac{11}{2} - frac{49}{5} cdot frac{28}{5} - frac{11}{2} cdot frac{28}{5} + frac{29}{10} cdot frac{79}{10}right)x - left(frac{49}{5} cdot frac{11}{2} cdot frac{28}{5}right) ] Simplifying the coefficients, we find: [ p(x) = x^3 - frac{62}{5}x^2 + frac{5191}{100}x - frac{41407}{100} ] Thus, the characteristic polynomial is: [ p(x) = x^3 - frac{62}{5}x^2 + frac{5191}{100}x - frac{41407}{100} ]