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f(1)  =  f(1- 1) + 2(1)  =  f(0) + 2  =  2

f(2)  = f(2 - 1) + 2(2)  =  f(1) + 4  =  2 + 4  =  6

f(3)  = f(3 - 1) + 2(3)  =  f(2) + 6  =  6 + 6  = 12

f(4)  = f(4 - 1) + 2(4) + f(3) + 8  =  12 + 8  =  20

Probability of 3 white balls matching =1 / 20C3 =1 / 1140.

Probability of the blue Superball =1 / 10C1 =1 / 10

Probability of of the winning ticket =1 / 1140 * 10 = 1 /11, 400.

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I think the smallest positive N = 98

98 / 7^2 = 2

N + 1 =99

99 / 3^2 =11

N + 2 = 100

100 / 5^2 =4

100 / 2^2 = 25

A line parallel to the given two lines ....and directly in the middle between them,

deleted.......k = pi    but I do not know how to derive it.....yet!

67/475

https://web2.0calc.com/questions/lottery-question_1

\(A)~P[\text{1 card of each suit$]= 1 \cdot \dfrac 3 4 \cdot \dfrac 1 2 \cdot \dfrac 1 4 = \dfrac{3}{32}$}\)

To save on desmos you have to creat an acount and creat the graph and click the green save button or you can go to the guid and select from there along with going back to the saved graphs

Travisio

11111111111111111111-2222222222 = 11111111108888888889

1 1  1 1 1  1 1 1  1 1 1   1 1 1  1 1 1  1 1 1

                            -   2   2 2 2   2 2 2  2 2 2

1 1  1 1 1  1 1 1  1 0 8  8 8 8  8 8 8   8 8 9

 Travisio

Problems posting   (internal server error)  ... corrected last step in answer below....   ~EP

find f^-1 =                y = x+1    solve for x and then switch x and y's     x = y-1     so  f^-1 = y = x-1

find g^-1                  y = 2x    solve for x   x = y/2    switch y's and x's    y = x/2  = g^-1

Now f(5) (this is the original f(x) function....not thte inverse)   f(5) = 5+1 = 6

  then  g(6) = 2x = 2(6) = 12

   then f^-1  (12) = 11

    then  f^-1 (11) = 10

      then g^-1 (10 ) = 5

       then f (5) = 6

find f^-1 =                y = x+1    solve for x and then switch x and y's     x = y-1     so  f^-1 = y = x-1

find g^-1                  y = 2x    solve for x   x = y/2    switch y's and x's    y = x/2  = g^-1

Now f(5) (this is the original f(x) function....not thte inverse)   f(5) = 5+1 = 6

  then  g(6) = 2x = 2(6) = 12

   then f^-1  (12) = 11

    then  f^-1 (11) = 10

      then g^-1 (10 ) = 5

       then f (7) = 6 

Mathematics is used for just about everything.

e.g.  Surveying and cartography is almost solely based on trigonometry.

Engineering is almost entirely maths and physics and the properties of materials.

Without engineering you would have no modern bridges or scyscrapers or airoplanes or just about anything else that makes up our modern existance.

If you want to know EXACTLY how then study hard and the facts will be revealed.

The expression p₁^Q₁*p₂^Q₂*....*p_n^Q_n for p₁,....,p_n distinct primes and Q₁,....,Q_n rational numbers will result in an integer if and only if Q₁,...,Q_n are all nonnegative integers

LHS:

\(\displaystyle 2^{x+y}.3^{x-y}.6^{2x+2y}=2^{x+y}.3^{x-y}.(2\times3)^{2x+2y}=2^{x+y}.3^{x-y}.2^{2x+2y}.3^{2x+2y}\\=2^{3x+3y}.3^{3x+y}\)

RHS:

\(\displaystyle 72=2^{3}.3^{2}\)

It's an identity, so equating indices,

\(\displaystyle 3x+3y=3,\\3x+y=2.\)

Solve, x = 1/2, y = 1/2.

The solution is unique.

i love desmos it's so user friendly and easy to usee

Thank you CPhill !

Ok, Thank you:

:)

You're exactly right!! 👍 🤪  🙌

And you can always check your answer to these type of questions with a graphing calculator such as desmos.