CPhill

(n - 2) (n + 4)  < 0

If we solved this

(n - 2) ( n + 4)  = 0

The solutions would be  n  = 2  and n  =  -4

So......the solutions to  the inequality lie on the possible intervals

(-inf, - 4) , ( -4, 2)  ,  (2, inf )

Testing  n  =  0  in the middle interval   will make the inequality true, while the outside two intervals don't 

So...the solution is  (-4, 2)   ....and  5 integers lie in this interval

You are correct, g(x)  =  x  / [ x + 2]

So

(g ° h) x  =     g ( h(x))   =   

[ (3x + 7) / (3x + 9) ]  /  [ (3x + 7) / (3x + 9) + 2  ]   =

[ (3x + 7)/ (3x+9) ]  /  [   ( 2 (3x + 9) + (3x + 7) )  /  (3x + 9) ]  =

[ (3x + 7)/ (3x + 9) ]  /  [ (9x + 25) / (3x + 9) ]  =

3x + 7               3x + 9

_______   *      _______

3x + 9               9x + 25

  3x + 7

_______

  9x + 25

 √ [ 2x - 3 ]  + 3  =  x        subtract 3 from both sides

 √ [ 2x - 3 ]  =  x - 3        square both sides

2x - 3   =  x^2 - 6x + 9       subtract/add   2x, 3  to both sides

0  = x^2 - 8x + 12      factor

(x - 6) ( x - 2)  = 0

Setting each factor to 0  and solving for x  gives x  = 6  and x  =  2

Only the first solution solves the original problem

Here's another way to do this

The  new volume will  be   (1/2)^3  =  1/8  of the original volume....so

904.32  * (1/8)   =  113.04  in^3

Third one

We have 

1 + 2 + 4 + .....+ 2^(n - 1)

The sum  is

1 ( 1 - 2^12)  / ( 1 - 2)   =    4095  cars

Last one

The total distance  traveled in the first 5 bounces is

80 + 80  ( 1 - (1/2)^(5 - 1) ) / ( 1 - 1/2)    = 230 ft

And the distance traveled from  the 5th bounce to the next peak is  80 (1/2)^5 =  2.5 ft

So...the total distance traveled is  230  + 2.5  =   232.5 ft

Note that when  all sides = 25.....the perimeter = 100  and the area  =  25^2  =  625 units^2

Note that when one side is 1  and the other is 49....the perimeter is

2(1 + 49)  = 2* 50  = 100

And the area is  1 * 49  =  49 units^2

And this is the least possible area for integer sides with a perimeter of 100

1.  -4 + 8 - 16 + 32 - 64 + 128 - 256   =

(-4 + 8)  + (-16 + 32) + (-64 + 128)  - 256  =

4  +  16  + 64 -  256  =

84  -  256  =

-172

Second one

4 + 12 +  36   +  108  +  .....+  8748

We need to find what term the last is

We can solve this

4*3^(n-1)  =  8748    divide both sides by 4

3^(n - 1)  =   2187      take the log of both sides

log 3^(n- 1)  =  log 2187     and we can write

(n - 1) log 3  = log 2187      divide both sides by log 3

(n - 1)  =   log 2187  / log 3

n - 1  =  7    add 1 to both sides

n  = 8

The sum is given by

4 ( 1 - 3^8) / ( 1 - 3)   = 13,120

Alan is correct....

if x  = 10  ⇒   (10 + 5) (10 - 5)   =  15 * 5   =  75

If x  =  -10  ⇒  (-10 + 5)  (-10 - 5)  =  -5 * -15   =  75

A recursive formula  can be used to find a term based upont the previous term

An algebraic formula can be used to find any term

First one

a_n  = 5  + 5(n - 1)    where n is the nth term

Second one  is a geometric series

a_n  =   3ⁿ     where n is the nth term

Last one

a_n   =       n 

             ____        where n is the nth term

              n + 1

(4 +  11i)  (4 - 11i)