CPhill

8 (1/2    - 1/4) > 12 - 2x

8 ( 2/4 - 1/4) > 12 - 2x

8 (1/4) > 12 - 2x

2 >  12 - 2x           add 2x to both sides, subtract 2 from both sides

2x >  10             divide both sides by 2

x > 5

Note that

3^2 - 1 =  8 = 2*4

5^2 - 1 = 24 = 4*6

7^2 - 1 = 48 =  6*8

So  we have  for n = 1 to infinity

6 *    1 / [ 2n * (2n + 2) ]  =  6 *   1 / [ 4 (n)(n+ 1) ]   =  (3/2) * 1/ [ n * (n + 1) ]

Note that     1  / [ n * (n + 1) ]   can be written as    1 /  n - 1/{n + 1]

So we have

(3/2)  [ ( 1 - 1/2 ) + (1/2 - 1/3) + (1/3 - 1/4) + (1/4 - 1/5) + ...... ]    

All the terms cancel  except  for the first term , 1,   and the last term , -1/(n + 1)

And when n is large  - 1 (n + 1) → 0

So....the sum is

(3/2) (1)  =

3/2

Notice that

1   + 2 - 3 = 0

4 + 5 -   6 = 3

7 + 8 - 9 =   6

So....the first 24 terms will be groups of 3 and there will be 8 of these groups 

And the last group will sum to  3(8-1) = 21

So the sum of these 8 groups will be    [ 0 + 21] * (8/2)  =  21 * 4 =  84

And the 25th term = 25

So...the sum is

84 + 25 =

109

OK...we got something we can work with...

If AFM is 60°.....then angle RFM =  30°

And the top left triangle is a 30-60-90 right triangle

And the side opposite the 90°  angle is the side of 7

And the side opposite the 30° angle is 1/2 the side length = 1/2*7 =  3.5

So... 1/2 of diagonal MA =  1/2(7) = 3.5

So....the whole length of AM is twice this = 7 units

And angle FMR is supplemental to angle AM =  120°

Then angle AMR is 1/2 of this = 60°

And the bottom left triangle is also 30-60-90

The side opposite  the 60° angle  = 3.5sqrt(3) units

And this is 1/2 the length  of diagonal  FR

So..its length = 2 *3.5sqrt(3)  =  7sqrt (3) units

3  4/9  =   31 / 9

Don't think we can solve this one without some other info, Josey......

See here :  https://web2.0calc.com/questions/algebra-help_37

sin T =   opp/ hyp = n / c

cos R =  adj / hyp =  n / c

So...equal!!!

Goodnight!!

1. The max angle is given by this :

arctan (1/12) = angle  ≈  4.76°

2. The tan is the rise over the run.....hence 1 ft rise / 12 foot run

The tan inverse (arctan) will tell us what an angle with this rise and run is 

3. Ramp length can be found by this

sin (4.76°) = 12 / ramp length        .....rearrange as

ramp length =    12/ sin (4.76°) ≈   144.6 ft

4. The sine gives the ratio of the opposite side from the angle to the hypotenuse which is the ramp length