CPhill

Thanks, PM...!!!

They are trying to be tricky here, RP

x = 8     is a vertical line through....you guessed it....x = 8

It will cover all y values...so the points (8,y) will be on this line

So a vertical line through x = 4  will be parallel to  x = 8

And it will cover all y values ....so the points  (4, y) will be on this line  ....so  ( 4, -2) is on the line 

So....the equation is

x = 4

And thanks to EP, too !!!

1/2 the length of each diagonal will form two legs of a right triangle

And the side of the rhombus will be the hypotenuse 

So....the perimeter is 4 times this hypotenuse length

4 sqrt ( 6^2 + 8^2) =

4 sqrt ( 36 + 64) =

4sqrt (100) =

4 * 10 =

40 units

OK, RP....you have already done this without realizing it...

Slope of given line = -4        point ( 1, -2)

y = slope ( x - x coordinate) + y coordinate

y = -4 ( x - 1) + (-2)

y = -4x + 4 - 2

y = -4x + 2

Show that is is true for  n = 1

7 (7¹ - 1)  = 7 (6) = 42      so...true !!!!

Assume it is true for n = k

That is :

7(7^k - 1)   is divisible by 42

Note that we can write this as 

(7* 7^k - 7)  

Prove it is true for n = k + 1

7(7^{k+ 1} - 1)  =

7 ( 7 * 7^k - ( 7 - 6) ) =

7 (  [7 * 7k - 7 ] + 6 ) =

7 ( 7 * 7^k - 7)  + 7*6

Note that the first term is divisible by 42 since ( 7 * 7^k - 7) was assumed to be divisible by 42

And 7*6 is clearly divisible by 42

So ...   7(7^{k+ 1} - 1) is divisible by 42

Here's (2)  using my method

m = (1/2)        point = ( 6, 1)

y = slope ( x -   x coordinate ) + y coordinate

y = (1/2) ( x - 6) + 1    simplfy

y = (1/2)x - 6/2 + 1

y = (1/2)x - 3 + 1

y = (1/2) x - 2 

So....Correct !!!

Thanks, EP !!!

Here's another way

(1)   m = 3            point = ( - 1, 4)

Write the equation as

y = slope ( x -   the x coordinate)   + the y coordinate     

So we have

y = 3 ( x -  -1) + 4

y  = 3 ( x + 1) + 4       simplify

y = 3x + 3 + 4

y = 3x + 7

OK....you have the correct slope for (5)  =  5/3

So...a perpendicular line will have the slope = -3/5

And an equation for a perpendicular line :   y = (-3/5)x

For (6)

4y - 1 = - 1x     add 1 to both sides

4y  = -1x + 1        divide both sides by 4

y = (-1/4)x + 1/4        the slope is (-1/4)

A perpendicular line has the slope =   4/1  = 4

( "Flip"  -1/4 = -4/1  ..   take the negative of this = - (- 4/1) = 4/1 = 4 )

So....the equation for a perpendicular line is  :    y = 4x

Hope that makes some sense.....!!!

u_n = 3 * 2^n-1 - 1     

Prove this is true for  n = 1

u₁ = 3 * 2 ^{1 - 1} + 1

u₁ = 3 * 1 - 1 =   2     which is true

Asume that it is true for n = k

That is : 

u_k = 3 * 2^k-1 - 1

Show that it is true for n = k + 1

u_k+1 = 3 * 2 ^{(k + 1 - 1)}  - 1

u_k+1 = 3 * 2^k - 1     (1)

And we were given  that

u_k+1  = 2u_k + 1        (2)

So....by our assumption u_k = 3 * 2^k-1 - 1    (3)

Sub    (3) into (2)

u_k+1  = 2 ( 3 * 2^k-1 - 1) + 1  =

2 * 3 * 2^k-1 - 2 + 1  =

3 * 2 * 2^k-1 - 1 =

3 * ( 2¹ * 2^k-1) - 1 =

3 * ( 2 ^{k - 1 + 1})  - 1   = 

3 * 2^k - 1        which   equals  ( 1)