Which of the following represents the union of set A and B? What are the first seven terms of the sequence defined by U1 = 0, U2 = 2, and Ur = Ur−1 − U_r-2? For the equation 3y2 - 2y extsqrt(6) + 2 =

Given $U_1 = 0$ and $U_2 = 2$.
The recurrence relation is $U_r = U_{r-1} - U_{r-2}$.

For $r=3$: $U_3 = U_2 - U_1 = 2 - 0 = 2$
For $r=4$: $U_4 = U_3 - U_2 = 2 - 2 = 0$
For $r=5$: $U_5 = U_4 - U_3 = 0 - 2 = -2$
For $r=6$: $U_6 = U_5 - U_4 = -2 - 0 = -2$
For $r=7$: $U_7 = U_6 - U_5 = -2 - (-2) = -2 + 2 = 0$

The first seven terms are $0, 2, 2, 0, -2, -2, 0$.
This matches option A.

The first seven terms are $\boxed{\text{A } 0,2,2,0,-2,-2,0}$.

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The union of set A and B is a fundamental concept in set theory. It represents a set containing all elements that are in A, or in B, or in both.

The standard notation for the union of set A and set B is $A \cup B$.
This matches option B.

The union of set A and B is $\boxed{\text{B } A \cup B}$.

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The given equation is $3y^2 - 2y\sqrt{6} + 2 = 0$.
This is in the form $ay^2 + by + c = 0$, where $a=3$, $b=-2\sqrt{6}$, and $c=2$.

The discriminant is given by the formula $\Delta = b^2 - 4ac$.

• If $\Delta > 0$, the roots are real and unequal.
• If $\Delta = 0$, the roots are real and equal.
• If $\Delta < 0$, the roots are complex (non-real).

Since $\Delta = 0$, the roots of the equation are real and equal.
This matches option A.

The nature of roots of the equation are $\boxed{\text{A Real and equal}}$.