answer:To find the equation of a circle passing through the points left(0,0right), left(4,0right), and left(-1,1right), we assume the general form of a circle's equation as x^{2}+y^{2}+Dx+Ey+F=0. Given these points, we can substitute them into the equation to form a system of equations to solve for D, E, and F. 1. Substituting left(0,0right) into the equation gives 0 + 0 + 0 + 0 + F = 0, which simplifies to F = 0. 2. Substituting left(4,0right) into the equation gives 16 + 0 + 4D + 0 + F = 0. Since we know F = 0, this simplifies to 16 + 4D = 0, which further simplifies to 4D = -16, and thus D = -4. 3. Substituting left(-1,1right) into the equation gives 1 + 1 - D + E + F = 0. Substituting D = -4 and F = 0 into this equation gives 2 + 4 + E = 0, which simplifies to E = -6. Therefore, the equation of the circle passing through the points left(0,0right), left(4,0right), and left(-1,1right) is x^{2}+y^{2}-4x-6y=0. Similarly, following the same process for the other sets of points, we find the equations of the circles as: - For points left(0,0right), left(4,0right), left(4,2right), the equation is x^{2}+y^{2}-4x-2y=0. - For points left(0,0right), left(-1,1right), left(4,2right), the equation is x^{2}+y^{2}-frac{8}{3}x-frac{14}{3}y=0. - For points left(4,0right), left(-1,1right), left(4,2right), the equation is x^{2}+y^{2}-frac{16}{5}x-2y-frac{16}{5}=0. Therefore, the possible equations of the circle passing through the given points are: boxed{x^{2}+y^{2}-4x-6y=0}, boxed{x^{2}+y^{2}-4x-2y=0}, boxed{x^{2}+y^{2}-frac{8}{3}x-frac{14}{3}y=0}, text{ and } boxed{x^{2}+y^{2}-frac{16}{5}x-2y-frac{16}{5}=0}.

answer:To find the number of ways to represent a number ( n ) as a sum of several integer terms ( a_i geq 2 ), where representations that differ by the order of terms are considered different, we can approach the problem as follows: 1. **Define the quantity to be discovered:** Let ( x_n ) be the number of ways to represent the number ( n ) under the given conditions. 2. **Base cases:** We need to determine the initial values for small ( n ): - For ( n = 1 ): There are no ways to write 1 as a sum of integers each greater than or equal to 2, thus ( x_1 = 0 ). - For ( n = 2 ): The only representation is ( 2 ) itself, thus ( x_2 = 1 ). - For ( n = 3 ): The only representation is ( 3 ) itself, thus ( x_3 = 1 ). 3. **Recursive relationship:** Now, we need to establish a relationship for ( x_n ) based on the previous values. For ( n geq 3 ), we consider the following: - If the first summand is ( 2 ), then the remaining terms must sum up to ( n - 2 ). The number of such representations is ( x_{n-2} ). - If the first summand is greater than ( 2 ), say ( k geq 3 ), then we can reduce ( k ) by 1 to count the number of representations of ( n - 1 ). The number of such representations is ( x_{n-1} ). Therefore, by considering both possibilities, the recursive formula is: [ x_n = x_{n-1} + x_{n-2} ] 4. **Conclusion:** The sequence ( {x_n} ) is the standard Fibonacci sequence shifted by one position. Thus, the solution can be represented as: [ x_n = F_{n-1} ] where ( F_n ) represents the ( n )-th Fibonacci number. Recalling that the Fibonacci sequence starts with ( F_1 = 1 ) and ( F_2 = 1 ), our final answer for the number of ways to represent ( n ) is: [ boxed{F_{n-1}} ]

answer:Let's denote the annual percentage increase in her expenditure as ( r ) percent. The expenditure after 1 year would be ( 1000 times (1 + frac{r}{100}) ). The expenditure after 2 years would be ( 1000 times (1 + frac{r}{100})^2 ). The expenditure after 3 years would be ( 1000 times (1 + frac{r}{100})^3 ). According to the given information, the expenditure after 3 years is Rs. 2197. So we can set up the equation: [ 1000 times (1 + frac{r}{100})^3 = 2197 ] To solve for ( r ), we first divide both sides by 1000: [ (1 + frac{r}{100})^3 = frac{2197}{1000} ] [ (1 + frac{r}{100})^3 = 2.197 ] Now we take the cube root of both sides: [ 1 + frac{r}{100} = sqrt[3]{2.197} ] Using a calculator, we find the cube root of 2.197: [ 1 + frac{r}{100} approx 1.3 ] Now we subtract 1 from both sides to find ( frac{r}{100} ): [ frac{r}{100} approx 1.3 - 1 ] [ frac{r}{100} approx 0.3 ] Finally, we multiply both sides by 100 to find ( r ): [ r approx 0.3 times 100 ] [ r approx 30 ] So the annual percentage increase in her expenditure is approximately boxed{30%} .

answer:Let's call the number of cupcakes Carol made initially "x." After selling 9 cupcakes, Carol had x - 9 cupcakes left. Then, she made 28 more cupcakes, so she had x - 9 + 28 cupcakes. According to the information given, after making the additional cupcakes, Carol had 49 cupcakes in total. So, we can set up the equation: x - 9 + 28 = 49 Now, let's solve for x: x + 19 = 49 Subtract 19 from both sides: x = 49 - 19 x = 30 Carol made boxed{30} cupcakes initially.