answer:Solution: (1) We calculate that a_1= frac{1}{2}; a_2= frac{1}{6}; a_3= frac{1}{12}; a_4= frac{1}{20}. (2) Conjecture: a_n= frac{1}{n(n+1)}. We will prove this using mathematical induction. (i) When n=1, the conjecture obviously holds. (ii) Assume when n=k (kinmathbb{N}^*), the conjecture holds, i.e., a_k= frac{1}{k(k+1)}. Then, when n=k+1, we have S_{k+1}=1-(k+1)a_{k+1}, which means S_k+a_{k+1}=1-(k+1)a_{k+1}. Since S_k=1-ktimes a_k= frac{k}{k+1}, we get frac{k}{k+1}+a_{k+1}=1-(k+1)a_{k+1}, thus a_{k+1}= frac{1}{(k+1)(k+2)}= frac{1}{(k+1)[(k+1)+1]}. This means the conjecture also holds when n=k+1. Therefore, by (i) and (ii), we know the conjecture is valid. Thus, the final answers are: (1) a_1= boxed{frac{1}{2}}, a_2= boxed{frac{1}{6}}, a_3= boxed{frac{1}{12}}, a_4= boxed{frac{1}{20}}. (2) The expression for a_n is boxed{frac{1}{n(n+1)}}.

answer:To evaluate each statement for correctness: - **Statement A: Equal angles are vertical angles.** - Vertical angles are indeed equal when two lines intersect each other. However, not all equal angles are necessarily vertical angles. Equal angles can also be corresponding or alternate angles when two lines are parallel and cut by a transversal. Therefore, the statement as presented is incorrect because it implies a one-to-one correspondence that does not exist. - **Statement B: When two lines are intersected by a third line, corresponding angles are equal.** - The statement lacks the crucial condition that the two lines must be parallel for the corresponding angles to be equal. When a transversal intersects two parallel lines, corresponding angles are equal. Without the parallel condition, this statement is misleading and incorrect. - **Statement C: In the same plane, there is only one line perpendicular to a given line passing through a point.** - This statement is true. Given a line in a plane, and a point either on the line or not, there is exactly one line through that point which is perpendicular to the initial line. This is a fundamental property of Euclidean geometry. - **Statement D: The complement of an acute angle may be equal to its supplement.** - By definition, the complement of an angle is what, when added to it, equals 90 degrees. The supplement is what, when added to the angle, equals 180 degrees. For an acute angle (less than 90 degrees), its complement and supplement cannot be equal because the sum of an angle and its complement is always less than the sum of the angle and its supplement. This statement is incorrect. Given the analysis above, the correct answer is: boxed{C}

answer:From the piecewise function expression, we have f(-2)=(frac{1}{2})^{-2}-2=4-2=2, f(2)=0, hence f(f(-2))=0, If xleqslant -1, from f(x)geqslant 2, we have (frac{1}{2})^{x}-2geqslant 2, which implies (frac{1}{2})^{x}geqslant 4, then 2^{-x}geqslant 4, Thus, -xgeqslant 2, which gives xleqslant -2, in this case xleqslant -2. If x > -1, from f(x)geqslant 2, we have (x-2)(|x|-1)geqslant 2, This simplifies to x|x|-x-2|x|geqslant 0, If xgeqslant 0, we have x^{2}-3xgeqslant 0, which gives xgeqslant 3 or xleqslant 0, in this case xgeqslant 3 or x=0, If x < 0, we have -x^{2}+xgeqslant 0, which simplifies to x^{2}-xleqslant 0, giving 0leqslant xleqslant 1, however, this case has no solution, Combining all cases, we have xgeqslant 3 or x=0, Therefore, the answer is: boxed{0}, boxed{xgeqslant 3 text{ or } x=0} To solve the first part, utilize the given piecewise function expression and substitution method. For the second part, discuss the possible ranges of x and solve the resulting inequalities. This problem primarily assesses the computation of function values, and the key to solving it lies in applying the substitution method and the approach of discussing cases separately based on the given piecewise function expression.

answer:(1) For the power function y=x^{-1}, the graph does not pass through (0, 0). Thus, proposition (1) is incorrect. because y = x^{-1} therefore y(0) text{ is undefined} This eliminates proposition (1). (2) The graph of the function y=x (a power function with n=1) is indeed a straight line. y = x^1 y = x Hence, the proposition (2) is incorrect. (3) When n=0, the function y=x^n simplifies to y=1, which is a horizontal line, not counting the point (0, 0) where x^n is undefined for n=0. y = x^0 therefore y = 1 text{ for all } x neq 0 Proposition (3) is therefore incorrect. (4) For the function y=x^2 (where n=2 > 0), the function is not increasing for all x since it is decreasing for x < 0. y = x^2 y' = 2x text{ which is negative for } x < 0 So, proposition (4) is incorrect. (5) The power function y=x^n when n<0 is indeed decreasing in the first quadrant as the value of x increases. As x becomes larger, the negative exponent causes the value of y to approach zero. because n < 0 therefore y = x^n text{ decreases as } x text{ increases for } x > 0 Therefore, proposition (5) is correct. [ boxed{text{The correct proposition number is: (5).}} ]