4,12,36,108,324,972,____ What is the next number in the pattern?

Mathematics

2 answers:

AysviL [449]3 years ago

8 0

2,916 is going to be your answer, they were all multiplied by 3 hope this helps!

Nikolay [14]3 years ago

8 0

U have a geometric sequence with a common ratio of 3. this means u r multiplying each number by 3 to find the next number.

4 * 3 = 12
12 * 3 = 36
36 * 3 = 108
108 * 3 = 324
324 * 3 = 972
972 * 3 = 2916 <== the next number

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Can you help me with some steps to approach this proof .

Elis [28]

To prove two sets are equal, you have to show they are both subsets of one another.

• X ∩ (⋃ ) = ⋃ {X ∩ S | S ∈ }

Let x ∈ X ∩ (⋃ ). Then x ∈ X and x ∈ ⋃ . The latter means that x ∈ S for an arbitrary set S ∈ . So x ∈ X and x ∈ S, meaning x ∈ X ∩ S. That is enough to say that x ∈ ⋃ {X ∩ S | S ∈ }. So X ∩ (⋃ ) ⊆ ⋃ {X ∩ S | S ∈ }.

For the other direction, the proof is essentially the reverse. Let x ∈ ⋃ {X ∩ S | S ∈ }. Then x ∈ X ∩ S for some S ∈ , so that x ∈ X and x ∈ S. Because x ∈ S and S ∈ , we have that x ∈ ⋃ , and so x ∈ X ∩ (⋃ ). So ⋃ {X ∩ S | S ∈ } ⊆ X ∩ (⋃ ).

QED

• X ∪ (⋂ ) = ⋂ {X ∪ S | S ∈ }

Let x ∈ X ∪ (⋂ ). Then x ∈ X or x ∈ ⋂ . If x ∈ X, we're done because that would guarantee x ∈ X ∪ S for any set S, and hence x would belong to the intersection. If x ∈ ⋂ , then x ∈ S for all S ∈ , so that x ∈ X ∪ S for all S, and hence x is in the intersection. Therefore X ∪ (⋂ ) ⊆ ⋂ {X ∪ S | S ∈ }.

For the opposite direction, let x ∈ ⋂ {X ∪ S | S ∈ }. Then x ∈ X ∪ S for all S ∈ . So x ∈ X or x ∈ S for all S. If x ∈ X, we're done. If x ∈ S for all S ∈ , then x ∈ ⋂ , and we're done. So ⋂ {X ∪ S | S ∈ } ⊆ X ∪ (⋂ ).

QED

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2х + 5y = 6 x+2y = -3 what is x?

likoan [24]

Answer:

x = -3

In-Depth Steps:

1.) Start with the bottom equation: $x+2y=-3$

2.) Set $x$ alone on one side of the equal sign. Your equation will look like $3+2y=x$ .

3.) Plug $3+2y$ in for $x$ in the first equation. You will get $2(3+2y)+5y=6$ .

4.) Distribute. Your equation now looks like this: $(6+4y)+5y=6$ .

5.) Combine like terms. You will get $6+9y=6$ .

6.) Subtract 6 from both sides. You get $9y=0$ . We can learn that Y=0.

7.) Go back to either equation and put 0 in for y.

8.) In the top equation, it would look like $2x=6$ and the bottom equation would look like $x=-3$ . We basically got our answer of $x = -3$ , but we will plug it into the $2x=6$ equation, and in-fact, we get x = -3.