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(a) The sum of the numbers in the first row is 1/(1 - a).

The sum of the numbers in the second row is ab/(1 - a).

The sum of the numbers in the third row is (a^2 b^2)/(1 - a).

So, the sum of the numbers in the rows form a geometric sequence, which adds up to

1/(1 - a) + ab/(1 - a) + (a^2 b^2)/(1 - a) + ...  = 1/((1 - a)(1 - ab)).

(b) Since the colors of the chessboard alternate white and black, the sum of the numbers on the black squares is equal to the sum of the numbers on the white squares, except for the numbers that are on every other white square, and every other black square.

The sum of the numbers that are on every other white square is a/((1 - a)(1 - ab)), and the sum of the numbers that are on every other black square is b/((1 - a)(1 - ab)), so to find the sum of the numbers on the black squares, we take half the difference, which gives us a sum of

1/((1 - a)(1 - ab)) + 1/2*a/((1 - a)(1 - ab)) + 1/2*b((1 - a)(1 - ab)) = (a + b + 2)/(2(1 - a)(1 - ab)).

1 Answer. Three digit numbers which leave theremainder 3 when divided by 5 are 103, 108, 113,......., 998.

that is wrong

hmm nvm. I already solved it. Sorry about that!

arc(FD)  =  2·76^o  =  152^o 

arc(GF)  =  2·40^o  =  80^o 

arc(GSD)  =  2·64^o  =  128^o 

arc(SD)  =  ½·arc(FD)  =  76^o 

arc(SG)  =  arc(GSD) - arc(SD)  =   ...

angle(SDG)  =  ½·arc(SG)  =  ...

1-kx=-3y

3y-kx=-1

From here we plug in the given point.

3y-kx=-1

3(-3)-k(4)=-1

-9-4k=-1

-4k=8

k=-2

1)  Find the midpoint of the line segment:  mdpt  =  (  (1 + 5)/2,  (3 + 7)/2  )  =  ( 3, 5 )

2)  Find the slope of the line segment:  m  =  (y₂ - y₁) / (x₂ - x₁)  =  (7 - 3) / (5 - 1)  =  4/4  =  1

3)  The slope of the perpendicular bisector will be the negative reciprocal of the slope found in 2).

     m  =  -1

4)  We have a point  (3, 5)  and a slope  -1,  so we can use the point-slope form:

      y - y₁  =  m(x - x₁)     --->     y - 5  =  -1(x - 3)

5)  Rewrite this equation:     y - 5  =  -1(x - 3)

                                             y - 5  =  -x + 3

                                                  y  =  -x + 8

                                             x + y  =  8 

i don't see the question.

Call the center of the half-circle  'O'

       the right-hand base corner of the triangle  'A'

       the point of tangency on the right side  'X'.

Because OX is perpendicular to AX, the angle(AXO) is a right angle.

Because the triangle is equilateral, the angle(XAO) is a 60^o angle.

This means that angle(XOA) is a 30^o angle.

OA has length 2, so side OX has length sqrt(3).

OA is a radius of the half-circle, so the diameter is  2·sqrt(3).

thx yall!!!

no problem! :)

Thank you so much!

hi otterstar!

so we know that \((-2)^4=16\) since -2*-2*-2*-2=16

since there are 4 roots, and we've already covered all the real ones, we have to look for complex numbers now.

we also know that \(i^4=1\), so that means that \((2i)^4=16\) since 2^4 already equals 16, and multiplying it by 1 will equal 16 too.

lastly, we know that \((-2i)^4=16\)  since we already know that -2^4 equals 16 (as we found earlier) and multiplying it by i^4=1 will equal 16 too. 

so the solutions are \(\boxed{2,-2,2i,-2i}\)

i hope this helped you! :)

Your number = 63

See answer # 11 here:   https://web2.0calc.com/questions/help-plzzzz_3

Step 1:  Find the slope of  3x + 4y  =  -14

                                                  4y  =  -3x - 14

                                                    y  =  (-3/4)x - 14/4

             The slope is  -3/4

Step 2:  Find the slope of the perpendicular line; it will be the negative reciprocal:  4/3

Step 3:  Use the point slope form to find the equation:  y - y₁  =  m(x - x₁)

                                                                                       y - 7    =  (4/3)(x + 5)

Step 4:  Write this equation in slope-intercept form:

                          y - 7  =  (4/3)(x + 5)

                          y - 7  =  (4/3)x + 20/3

                               y  =  (4/3)x + 41/3

Step 5:  Identify  m  and  b.

Step 6:  Find  m + b.

The easiest way would be to draw an equilateral triangle 3 units on each side.

Call the original triangle ABC. 

Divide each side into thirds.

Label the two points of trisection of AB as U and V.

Label the two points of trisection of BC as W and X.

Label the two points of trisection of CA as Y and Z.

Draw VW, XY and ZU to get hexagon UVWXYZ.

Draw UX, VY, and WZ.

They will meet at O, the center of the triangle.

You now have 9 congruent trianges, 6 in the hexagon.

The ratio of the hexagon to the area of the original triangle:  6 : 9  =  2 : 3.

You can also find the areas of the triangles and get the same answer that way.

Start listing them (have the second digit larger than the first and the third digit larger than the second):

123     126     129     135     138     147     156     159     168

234     237     246     249     258     267     279

345     348     357     369     378    

456     459     468    489

567     579

678

789

But each of these can be chosen in 6 ways:   123  =  132  =  213  =  231  =  312  =  321

So take the number of ways listed and multiply by 6.

It would also be wise to check to see if I have missed any.

Sorry, but it sayd that thats wrong.

Make a vertical cut, dividing the object into a taller left section (12 x 5 x 6  =  360)

and a shorter right section (10 x 8 x 6  =  480)

and add these two volumes together.

oh wow! yay! :D

64  =  8 x 8

64  =  4 x 4 x 4

As the second guest states:  the answer is 9.6.

Here is an explanation:  XY · XY  =  XB · XA   (This is the correct example of "power of a point".)

                                            9 · 9  =  XB · 15

                                                81  =  15·XB

                                                XB  =  5.4

Since   XB + BA  =  XA

            5.4 + BA  =  15

                     BA  =  9.6