CPhill

Circumference  =   2pi * r  ....so....

4pi^2  = 2 pi * r        divide both sides by  2 pi

2 pi  =  r

So...the area  =

pi * r^2  =

pi * (2 pi)^2

4 pi ^3    units^2

  The square  triangular  numbers  , after 1, are

So   [ C(9, 7)]  =    9! / [ 2! * 7! ] =  (9 * 8) / 2  =   9* 4  = 36   

The next number occurs at   C(50,48)  =  (50!) /[ 48! * 2! ]  =  [50* 49] / 2 = 25*49  = 1225

3.Find the value of n that satisfies   \(\frac{1}{n+1} + \frac{2}{n+1} + \frac{n}{n+1} = 3\)

[ 1 + 2 + n ]

__________   =    3                   multiply through by n + 1

     n + 1 

3 + n  =  3n + 3      subtract n, 3 from both sides

0 = 2n   

n = 0

2. What is the y-intercept of the line x-2y=5?

The y intercept occurs where x = 0......so....

(0) - 2y = 5

-2y = 5        divide both sides by  -2

y = -5/2   = the y intercept

1.   4x + ay  = -8  (1)

      2x + y  = b           multiply through by 2  ⇒  4x + 2y = 2b     (2)

This will have infinite solutions  when (1) and (2) are the same

So    a  = 2

And  -8 = 2b  ⇒  b = -4

So  (a, b)  =  ( 2, - 4)

Hey, Manuel!!!

A concentric circle that has a radius greater than the given circle

P(A l B)  =       .24                                                  P( B l A)  = .24

                _____________  =   4/7  = .57                                ____  =   3/5  = .60

                       .42                                                                      .40

P(A)  = .40

P(B) = .42

Not independent because   

P(A l B)  not equal P(A)     and P(B l A)  not equal to B

They travel for 3 hours.....and the total distance covered = 240 miles

Let x be the rate of the slower car    and x + 10  be the rate of the faster

And    R*T  = D    so

3(x) + 3(x + 10) = 240

3x +  (3x + 30)  = 240

5^n + 6*7^n+ 1

Show that it's true for  n = 1

5^1 + 6*7^1 + 1  =

5 + 42 + 1  =  48     which is divisible by 8

Assume that it's true for  k ...that is

5^(k) + 6*7^k + 1     is divisible by 8

Prove that it is true for k + 1

5^(k + 1)  + 6*7^(k + 1) + 1

5*5^k + 7*6*7^k + 1

5*5^k + 42* 7^k + 1

5^k + 4*5^k + 6*7^k + 36*7^k + 1

(5^k + 6*7^k + 1)  +   4*5^k + 36*7^k

(5^k + 6*7^k + 1) +  4*(5^k + 9*7^k)

Note that  5^k  is odd   and so is 9*7^k  ....but the sum of two odds is even = 2n

So we have

(5^k + 6*7^k + 1) + 4 (2n)  =

(5^k + 6*7^k + 1)  + 8n

The first was assumed to be divisible by 8   and the second term is also divisible by 8

I think I like your method better, hectictar  ....!!!