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3 years ago

Prove this conjecture

Mathematics

1 answer:

steposvetlana [31]3 years ago

7 0

Answer:

See explanation

Step-by-step explanation:

Consider the natural number $n(n+1)(n+2).$

If $n$ is the natural number, then $n+1, \ n+2$ are next two natural numbers, so numbers $n,\ n+1$ and $n+2$ are three consecutive numbers.

Given three consecutive natural numbers, one of them is always even number and one of them is always multiple of 3, so the product

$n(n+1)(n+2)$

is divisible by 6.

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The midpoint of segment AB is (4, 2). The coordinates of point A are (2, 7). Find the coordinates of point B.

erastovalidia [21]

Answer:

<h3>The option B) is correct</h3><h3>Therefore the coordinate of B $(x_2,y_2)$ is (6,-3)</h3>

Step-by-step explanation:

Given that the midpoint of segment AB is (4, 2). The coordinates of point A is (2, 7).

<h3>To Find the coordinates of point B:</h3>

Let the coordinate of A be $(x_1,y_1)$ is (2,7) respectively
Let the coordinate of B be $(x_2,y_2)$
And Let M(x,y) be the mid point of line segment AB is (4,2) respectively
The mid-point formula is

Now substitute the coordinates int he above formula we get
$(4,2)=(\frac{2+x_2}{2},\frac{7+y_2}{2})$
Now equating we get

$4=\frac{2+x_2}{2}$ $2=\frac{7+y_2}{2}$

Multiply by 2 we get Multiply by 2 we get

$4(2)=2+x_2$ $2(2)=7+y_2$

$8=2+x_2$ $4=7+y_2$

Subtracting 2 on both

the sides Subtracting 7 on both the sides

$8-2=2+x_2-2$ $4-7=7+y_2-7$

$6=x_2$ $-3=y_2$

Rewritting the above equation Rewritting the equation

$x_2=6$ $y_2=-3$

<h3>Therefore the coordinate of B $(x_2,y_2)$ is (6,-3)</h3><h3>Therefore the option B) is correct.</h3>

7 0

3 years ago

Find the value of the variable and the measure of each labeled angle.

nalin [4]

X + 18 = 5x - 38

x - x + 18 = 5x - x - 38

18 = 4x - 38

18 + 38 = 4x - 38 + 38

56 = 4x

56/4 = 4x/4

14 = x

3 0

3 years ago

Plz help I don't know this answer

tester [92]

Answer:

4 passed 1 failed

Step-by-step explanation:

7 0

3 years ago

Read 2 more answers

Simplify: -4g(8g + 5)

VMariaS [17]

Answer:

−32^2−20

Step-by-step explanation:

4 0

2 years ago

Find the flux of F=(x^5+y^5+z^5-2x-3y-4z)i+sin(2y)j+4zsin^2(y)k across the surface of the tetrahedron bounded by the coordinate

Tju [1.3M]

Use the divergence theorem.

$\mathbf F(x,y,z)=(x^5+y^5+z^5-2x-3y-4z)\,\mathbf i+\sin2y\,\mathbf j+4z\sin^2y\,\mathbf k$
$\implies(\nabla\cdot\mathbf F)(x,y,z)=\dfrac{\partial(x^5+y^5+z^5-2x-3y-4z)}{\partial x}+\dfrac{\partial(\sin2y)}{\partial y}+\dfrac{\partial(4z\sin^2y)}{\partial z}=5x^4-2+2\cos2y+4\sin^2y$

The flux of $\mathbf F$ across the tetrahedron's surface $S$ is then given by the integral of $\nabla\cdot\mathbf F$ over the interior of the tetrahedron $\mathbf R$ .

$\displaystyle\iint_S\mathbf F\cdot\mathrm dS=\iiint_T\nabla\cdot\mathbf F\,\mathrm dV$
$=\displaystyle\int_{x=0}^{x=1}\int_{y=0}^{y=1-x}\int_{z=0}^{z=1-x-y}(5x^4-2+2\cos2y+4\sin^2y)\,\mathrm dz\,\mathrm dy\,\mathrm dx$
$=\dfrac1{42}$

4 0

3 years ago