CPhill

Recursive formula

a_{n + 1}  =   a_n  +  4             where a₁  =  10

Finite series

(1) 1, 2, 3, 4

The  infinite series  is  represented  by   ....     after the last listed term

(3)  is just a sum

(4)  is an infinite sum

g(f(x))  just  means  that  we are putting   "f"  into each  "x"  in  g   and evaluating

So......

  ( (x-2)  / ( x + 3) ) -  4  =

(x - 2)  - 4(x + 3)

_____________   =

      x + 3

-3x  - 14

_______

   x + 3

f(g(x))  (as you may  have guessed)  means  "g"  into "'f

So....

(x- 4)  - 2

_________   =

(x - 4) + 3

x - 6

_____

 x - 1

tanx  + cotx

____________

csc x  cos x

sinx/cos x  + cos x / sin x

____________________

   1/sinx *  cos x

sin^2 x  + cos ^2x

______________

      sinx cos x

_______________

        cosx / sin x

           1

____________

 sin x cos x

________________

    cos x / sin x

          1                     sin x

____________*     ______

   sin x cos x             cos x

             1

________________

         cos ^2 x

        sec^2  x

First one

When 6 + 4i  is multiplied by 3, the result is 18 + 12i. Graphically, this shows that the product is a scalar of the complex number.

Second one

Converges     (gets smaller and smaller)

Geometric  and the absolute value of the common ratio is less than 1    (=2/3)

Last one

The limit is just  the ratio of the  coefficients on the   n^4  terms  in the numerator and denominator = 2/6  = 1/3

 a. Express the sequence, a_n, using an explicit formula.

a_n  =  5* (3)^(n - 1)    where  n ≥  1

b.  16th term  

a₁₆ =   5*(3)^15  =  71744535

c. Recursive formula

a_n+1  =   3a_n                 where  a₁   =   5

6^3  =  216  Possible Outcomes

1 way for   3

3 ways for 4

6 for 5

10 for 6

15 for 7

21 for 8

25 for 9

27 for 10

27 for 11

25 for 12

21 for 13

15 for 14

10 for 15

6 for 16

3 for 17

1 for 18

Multiples  of  4  =   4 , 8 , 12 , 16

So

( 3  +  21  +  25  + 6) /   216   =

  55  / 216

We  have  this system

100 T  +  94 PB  = 776             (1)

85 T  +   94 PB   =  716   ⇒  -85 T   -  94 PB   =  - 716       (2)

Add  (1)  and (2)

15 T  =  60

T =  60 / 15    =   $4  a box for truffles

And

100 (4)  +  94 PB   =  776

400  +  94 PB  =  776

94 PB  =  776 - 400

94 PB  =  376

PB  =  376  / 94    =  $4  a box for peanut brittle

(10, 8)      ( -15 , 18)

Slope   =     ( subtract  y coordinates in some order)  /  ( subtract x coordinates in the  same order ) =

So....

( 18 - 8 )   /  ( -15 -10)  =  

10 /  -25   =

-2 / 5

NP !!!!