CPhill

17,755,477,825,681,042,807,338

17 sextillion 755 quintillion 477 quadrillion 825 trillion 681 billion 42 million 807 thousand 338

And that's a mouthful....!!!

We got bored with just staring at a calculator all the time....so.....it just seemed like the thing to do.....

Correct, NSS    ....

8 < 4x + 4 < 2(x + 10)

We have two inequalities to consider

8 < 4x+ 4                                                   4x +4  < 2(x + 10)

Subtract 4 from both sides                        divide through by 2

4 < 4x                                                         2x + 2 < x + 10

Divide both sides by 4                               Subtract 2, x from both sides

1 < x    (1)                                                           x  < 8   (2)

Combining (1) and (2) we have

1 < x < 8

Could you give an example????

Whoa!!!......very well done......that was a difficult one!!!

Nice, heureka !!!

We want all the sets of any 3 elements chosen from 8....and then we want alll the possible orderings of those sets....so

P (8,3)  =   336

Others on here can probably do this in a more elegant manner, MC, but.....here's my clumsy attempt

n^2 = 5p + 4

n^3 = 5q + 2

Subtract the first equation from the second and rearrange as

5 (q - p) = [ n^3 - n^2 + 2 ]

q - p =  [ n^3 - n^2 + 2] / 5

Note that the right side must be divisible by 5

So....the first "n" I find that works is  n = 3

So....... 20/ 5 = 4         and q - p = 4

So  

3^2 = 5(1) + 4

3^3 = 5(5) + 2

So.....  n / 5  =  3 / 5  =   3 mod 5  = 3 = Remainder

a)   f ' (2) =  20(2) - 6(2)^2 - 16  =  40 - 24 - 16 = 0

= the slope of the x axis......so.....parallel

The equation of this tangent line is y = 0 (x -2) + 3 .... y = 3         (1)

b)  The slope of the tangent line at x = 3 is

20(3) - 6(3)^2 - 16 =  60 - 54 - 16 =  -10

So....the slope of the normal line is (1/10)

The equation of this normal line is

y = (1/10)(x - 3) -1

y = (1/10)x - 3/10 - 1

y = (1/10)x - 13/10          sub (1) into this to find the x coordinate of R

3 = (1/10)x - 13/10

3 + 13/10 = x /10

43/10 = x /10

43 = x

So.....R = (43, 3)

See the graph here to confirm this :

https://www.desmos.com/calculator/0ipyvcpc5y

P.S. - If you have had Integral Calculus....you might see how I determined f(x)