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Oh thanks, I posted this question because it wouldn't work on this website. Hadn't thought of using other websites, thanks :D

The greatest integer on the "odd" side of the street is

3 + 6(20 - 1) =  3 +  6(19) = 3 + 114 = 117

2 houses have one digit each 3, 9

3 houses will have three digits each 105, 111, 117

So...15 houses will have two digits each 

So...for the "odd" side he collects  2(1) + 15(2) + 3(3) = 2 + 30 + 9 = $ 41   

For the "even" side, the greatest integer is   4 + 6(20-1) = 4 + 6(19) = 118

1 house has one digit

4 houses have three digits each [ 100, 106, 112, 118]

So.....15 houses wil have two digits each

So....for the "even" side he collects 1(1) + 15(2) + 4(3) = 1 + 30 + 12 = $43

So....the total he collects is    $ [ 41 + 43 ]  =  $ 84

This is an exponential function.......

Here is the graph.....https://www.desmos.com/calculator/0vbgmdeqfv

Nicely done, Melody !!!!

Using the answer to your other question, the equation of the hyperbola was found to be

(y - 1)^2      -   (x - 0)^2

________      ________ =   1

      4                 1

The center is   (0, 1)  = (h, k)

a = sqrt (4)  =  2  

The vertices  are given by

(h, k+a)   and  (h, k - a) =

(0, 1 + 2)  and   (0, 1 - 2) =

(0, 3)   and (0, -1)

-4x^2 +y^2 -2y = 3      rewrite as

y^2 - 2y - 4x^2 = 3        complete the square on y

y^2 - 2y + 1 - 4x^2 = 3 + 1

(y - 1)^2  -   4x^2 =   4         divide both sides by 4

(y - 1)^2     -   x^2  

______         ____  =      1

   4                  1

In this form,   a = 2  , b = 1     h = 0    k = 1

The equation of the asymptote with the positive slope is 

y =  (a/b) ( x - h) + k        ....so...

y = (2/1) ( x - 0) + 1

y = 2x + 1

Here's the graph :   https://www.desmos.com/calculator/tenipf6zfq

Let the grid be numbered

123

456

789

Based on the symmetry lines the groups that must be the same color are

{5}

{1,3,7,9}

{2,4,6,8}

Thus 3 colors are the maximum that can be used.

A must be a right angle

AB = 1

AC = sqrt (3)

BC = 2

Let A = (0,0)  B = (1, 0)  and C = (0, sqrt (3) )

The midpoint of BC is  [ (1 + 0)/2 , ( sqrt(3) + 0 ) / 2 ]  =   (1/2, sqrt (3)/2 )

Call this point, D

The distance from A to D is

sqrt [   (1/2)^2 + (sqrt (3)/ 2)^2]  =  sqrt [  1/4 + 3/4] = sqrt (1) = 1

And the distance from A to G is 2/3 of AD =   2/3 units

this has been answered multiple times on this site.

use the search function

\(\dbinom{7}{3}\dbinom{4}{2} = 35 \cdot 6 = 210\)

No prob.....!!!!

Using Pascal's Triangle

C( n , k) + C(n, k + 1) = C(n+1, k + 1)

So

C(26, 13) + C(26, 14) = C( 27, 14)

But ....C(27, 14)  = C(27, 13)     ....so we also have

C(26, 12) + C(26, 13)  =  C(27, 13)

So

n = 12    or n = 14

And their sum =  26

well it's clear that among your x's 3 must be even and 3 must be odd.

The evens can occupy either slots 1,3,5 or slots 2,4,6

There are 3! = 6 ways you can arrange them in those slots.

The odds will occupy the remaining slots and again there are 6 ways to arrange them.

So in total there are 6*6=36 ways to arrange your x's as described.

thank you for your help!

< 58.5        < 63.5           < 68.5

     44             77                80

Take a subsequence of N terms of this.

Split each term up using partial fractions

You'll see this reveals that the sequence is a telescoping sequence.

After the sum only part of the Nth term remains

\(\dfrac{N}{b (a N+b (-N)+b)}\)

Taking the limit of this as N goes to infinity results in

\(\dfrac{1}{b(a-b)}\)

133/150

\(3/32\)

If 4 + i    a root then so is the conjugate 4 - i

Unfortunately.....We have to multiply these....!!!!

Multiplying these we get the polynomial

(x - (4 + i) ) ( x - (4 - i) )    =  x^2  - x(4 + i) - x(4 - i) + (4 + i)(4-i) =

x^2 - 8x + 16 - i^2 =

x^2 - 8x + 17

To find the remaining polynomial....we can perform some polynomial division

                         x + 3

x^2 - 8x + 17  [  x^3 - 5x^2 - 7x + 51 ] 

                         x^3 - 8x^2 + 17x

                        ___________________

                                 3x^2 - 24x + 51

                                 3x^2 - 24x + 51

                                ______________

Since x + 3  is the remaining polynomial, x = -3  is the remaining root

Solve for x:
x^3 - 5 x^2 - 7 x + 51 = 0

The left hand side factors into a product with two terms:
(x + 3) (x^2 - 8 x + 17) = 0

Split into two equations:
x + 3 = 0 or x^2 - 8 x + 17 = 0

Subtract 3 from both sides:
x = -3 or x^2 - 8 x + 17 = 0

Subtract 17 from both sides:
x = -3 or x^2 - 8 x = -17

Add 16 to both sides:
x = -3 or x^2 - 8 x + 16 = -1

Write the left hand side as a square:
x = -3 or (x - 4)^2 = -1

Take the square root of both sides:
x = -3 or x - 4 = i or x - 4 = -i

Add 4 to both sides:
x = -3 or x = 4 + i or x - 4 = -i

Add 4 to both sides:

x = -3          or x = 4 + i          or x = 4 - i

Answered here, too :

https://web2.0calc.com/questions/help-urgent_19

The smallest positive n = 3

3 mod 5 = 3 - the remainder.

The general solution for n is:

n = 5m + 3,  where m =0, 1, 2, 3.....etc.

The number of  data values  is   [ 10 * 11] / 2  =  110/2 = 55

The mean is

[1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 + 7^2 + + 8^2 + 9^2 + 10^2] / 55  =   7

The mode is the data value that appears most often =  10

The median is  also  7 

Look at the display ABOVE the calculator.....fractions are displayed there...