answer:To find the value of 2 diamondsuit 7, I need to plug in x = 2 and y = 7 into the formula and simplify. So, 2 diamondsuit 7 = 3(2) + 5(7). That gives me 2 diamondsuit 7 = 6 + 35.

answer:This is a geometric sequence, where each term is obtained by multiplying the previous term by a constant ratio. I can find the ratio by dividing any term by the previous term. For example, frac{512}{2048} = frac 14, so the ratio is frac 14. This means that to get the next term, I multiply the current term by frac 14. So, to find x, I multiply 128 by frac 14, and get x = 32. To find y, I multiply x by frac 14, and get y = 8. Therefore, x + y = 32 + 8 = 40.

answer:Since a is inversely proportional to b, I know that there is some constant k such that a=frac{k}{b} for any nonzero values of a and b. Therefore, I can write a_1=frac{k}{b_1} and a_2=frac{k}{b_2}. I also know that frac{a_1}{a_2}=frac{2}{3}, so I can substitute the expressions for a_1 and a_2 and get frac{k}{b_1}cdotfrac{b_2}{k}=frac{2}{3}. Simplifying this equation, I get frac{b_2}{b_1}=frac{2}{3}. Taking the reciprocal of both sides, I get frac{b_1}{b_2}=frac{3}{2}.

answer:To find the center of the circle, I need to complete the square for both x and y terms. I can rewrite the equation as (x^2+2x)+(y^2+10y)=-16. To complete the square for (x^2+2x), I need to add (2/2)^2=1 to both sides. To complete the square for (y^2+10y), I need to add (10/2)^2=25 to both sides. So I have (x^2+2x+1)+(y^2+10y+25)=-16+1+25. Simplifying, I get (x+1)^2+(y+5)^2=10. Now I can see that the center of the circle is (-1,-5), since the equation is in the form (x-h)^2+(y-k)^2=r^2, where (h,k) is the center and r is the radius. Therefore, x+y=(-1)+(-5)=-6.