answer:(1) From S_n + 2n = 2a_n, we can rewrite the relation as S_n = 2a_n - 2n. qquad text{(1)} When n is a positive integer, we have S_n = 2a_n - 2n. For n=1, we get S_1 = 2a_1 - 2, which implies a_1 = S_1 + 2 = 2. Then, for n geq 2, n is a positive integer, we have S_{n-1} = 2a_{n-1} - 2(n-1). qquad text{(2)} Subtracting (2) from (1), we get a_n = 2a_{n-1} + 2. Hence, a_{n+2} = 2(a_{n+2-1} + 2). Therefore, the ratio of consecutive terms is frac{a_{n+2}}{a_{n+1}} = 2, so the sequence {a_{n+2}} is a geometric sequence with the first term a_1 + 2 and common ratio 2. This means we have a_{n+2} = 4 cdot 2^{n-1}, and therefore the general formula for a_n is a_n = 2^{n+1} - 2. boxed{a_n = 2^{n+1} - 2} is the general formula for the sequence {a_n}. (2) Using b_n = log_2(a_n + 2) = log_2 2^{n+1} = n+1, we get frac{b_n}{a_n + 2} = frac{n + 1}{2^{n+1}}. Thus, the sum T_n is T_n = frac{2}{2^2} + frac{3}{2^3} + ldots + frac{n+1}{2^{n+1}}. qquad text{(3)} Considering frac{1}{2}T_n, we have frac{1}{2}T_n = frac{2}{2^3} + frac{3}{2^4} + ldots + frac{n}{2^{n+1}} + frac{n+1}{2^{n+2}}. qquad text{(4)} Subtracting (4) from (3), we obtain frac{1}{2}T_n = frac{2}{2^2} + left(frac{1}{2^3} + frac{1}{2^4} + ldots + frac{1}{2^{n+1}}right) - frac{n+1}{2^{n+2}}. This can be simplified using the sum of a geometric series to frac{1}{2}T_n = frac{1}{4} + frac{frac{1}{4}(1 - frac{1}{2^n})}{1 - frac{1}{2}} - frac{n+1}{2^{n+2}}. Simplifying further, we find frac{1}{2}T_n = frac{1}{4} + frac{1}{2} - frac{1}{2^{n+1}} - frac{n+1}{2^{n+2}}. Thus, we have frac{1}{2}T_n = frac{3}{4} - frac{n+3}{2^{n+2}}. Multiplying both sides by 2, we have T_n = frac{3}{2} - frac{n+3}{2^{n+1}}. Since the subtracted term frac{n+3}{2^{n+1}} is always positive for n geq 1, it ensures that boxed{T_n < frac{3}{2}} as required.

answer:For the hyperbola dfrac {x^{2}}{m}- dfrac {y^{2}}{8}=1 (where m > 0), we have a= sqrt {m}, b=2 sqrt {2}, and c= sqrt {a^{2}+b^{2}}= sqrt {8+m}, Given the eccentricity e= dfrac {c}{a}= dfrac { sqrt {8+m}}{ sqrt {m}}= sqrt {5}, solving this equation yields m=2. Therefore, the answer is boxed{2}. To determine the value of m for the hyperbola, we first ensure m > 0, then calculate a, b, and c. Using the formula for eccentricity e= dfrac {c}{a}, we establish an equation and solve it to find the value of m. This question tests knowledge of the equation and properties of a hyperbola, focusing on eccentricity, and involves equation-solving skills. It is considered a basic question.

answer:# Problem: Let ( f: mathbb{R} to mathbb{R} ) be a function such that for any numbers ( x ) and ( y ), [ |f(x) - f(y)| = |x - y|. ] What is the possible value of ( f(2) ) given that ( f(1) = 3 )? 1. **Substitute specific values:** - Since the equation holds for all ( x ) and ( y ), substitute ( x = 1 ) and ( y = 2 ): [ |f(1) - f(2)| = |1 - 2| = 1. ] - Given ( f(1) = 3 ), substituting into the equation gives: [ |3 - f(2)| = 1. ] 2. **Solve for ( f(2) ):** - We now have the absolute value equation: [ |3 - f(2)| = 1. ] - There are two possible solutions to this equation: [ 3 - f(2) = 1 quad text{or} quad 3 - f(2) = -1. ] - Solving these equations individually: [ begin{aligned} 3 - f(2) = 1 & implies f(2) = 2, 3 - f(2) = -1 & implies f(2) = 4. end{aligned} ] 3. **Verification with function forms:** - To better understand the function, let's investigate the general form of ( f ) consistent with ( |f(x) - f(y)| = |x - y| ): - **Case 1:** ( f(x) = x + c ): - If ( c = 2 ): [ f(x) = x + 2. ] - Check: [ |f(x) - f(y)| = |(x + 2) - (y + 2)| = |x - y|. ] - This form satisfies the given condition. - **Case 2:** ( f(x) = k - x ): - If ( k = 4 ): [ f(x) = -x + 4. ] - Check: [ |f(x) - f(y)| = |-x + 4 - (-y + 4)| = |-x + y| = |x - y|. ] - This form also satisfies the given condition. - Therefore, ( f(x) = x + 2 ) and ( f(x) = -x + 4 ) are valid functions that fit the requirement. 4. **Conclusion:** - The possible values of ( f(2) ) are 2 and 4. Thus, [ boxed{2 text{ or } 4}. ]

answer:For a sphere, the volume is frac{4}{3}pi R^3, where R is its radius. For a hemisphere, the volume formula is frac{2}{3}pi r^3, where r is its radius. Given that the volumes are equal when the sphere becomes a hemisphere, we equate the two volumes: [ frac{4}{3}pi R^3 = frac{2}{3}pi r^3 ] Simplifying by eliminating common terms and solving for R, we get: [ 2R^3 = r^3 Rightarrow R^3 = frac{1}{2}r^3 Rightarrow R = frac{1}{sqrt[3]{2}}r ] Plugging in the value of r = 5sqrt[3]{2}: [ R = frac{1}{sqrt[3]{2}} times 5sqrt[3]{2} = 5 ] Thus, the radius of the original balloon is boxed{5 text{ cm}}.