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wow..good luck bro

could you give the initial value?

Try the "continued fraction" method as follows:

0.140376266280752532561505065123............

Take its reciprocal:

7.123711340206185567010309278351............

Subtract 7 and take the reciprocal of the fraction:

8.08333333333333333333333333....................

Subtract 8 and take the reciprocal of the fraction:

And it converges to 12.  Take this number, 12, and multiply it by the number above it:

12 x 8.0833333333333333333...... = 97. This will be the numerator:

Multiply 97 x 7.123711340206185567010309278351

= 691 - This is the denominator. So, in this particular case, the decimal appears to be rational:

97 / 691 - Try dividing it and you will get this:

0.1403762662 8075253256 1505065123 0101302460 2026049204 0520984081 0419681620 8393632416 7872648335 7452966714 9059334298 1186685962 3733719247 4674384949 3487698986 9753979739 5079594790 1591895803 1837916063 6758321273 5166425470 3328509406 6570188133 1403762662.........etc. Its "period", or when the digits start repeating is 230 digits!.

Find  \(a+b+c\) , given that  \(x+y\neq -1\)  and  \( \begin{align*} ax+by+c&=x+7,\\ a+bx+cy&=2x+6y,\\ ay+b+cx&=4x+y. \end{align*}\)

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(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (ax + by + c) + (a + bx + cy) + (ay + b + cx)

(ax + by + c) + (a + bx + cy) + (ay + b + cx)  =  (x + 7) + (2x + 6y) + (4x + y)

ax + by + c + a + bx + cy + ay + b + cx  =  x + 7 + 2x + 6y + 4x + y

ax + bx + cx + ay + by + cy + a + b + c  =  7x + 7y + 7

x(a + b + c) + y(a + b + c) + 1(a + b + c)  =  7(x + y + 1)

(a + b + c)(x + y + 1)  =  7( x + y + 1)

a + b + c  =  7

I like math!

Simplify: Sorry Adam, no LaTex !!
sin(x)^4 - sin(x)^2 cos(x)^2 - 2 cos(x)^4 = 

Express sin(x)^2 in terms of cosine via the Pythagorean identity.
sin(x)^2 = 1 - cos(x)^2:
-2 cos(x)^4 - cos(x)^2 1 - cos(x)^2 + sin(x)^4 =

Expand -cos(x)^2 (1 - cos(x)^2).
-cos(x)^2 (1 - cos(x)^2) = cos(x)^4 - cos(x)^2:
-2 cos(x)^4 + cos(x)^4 - cos(x)^2 + sin(x)^4 =

Express sin(x)^4 in terms of cosine via the Pythagorean identity.
sin(x)^4 = (sin(x)^2)^2 = (1 - cos(x)^2)^2:
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + (1 - cos(x)^2)^2 =

Expand (1 - cos(x)^2)^2.
(1 - cos(x)^2)^2 = 1 - 2 cos(x)^2 + cos(x)^4:
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4 =

Evaluate -2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4.
-2 cos(x)^4 - cos(x)^2 + cos(x)^4 + 1 - 2 cos(x)^2 + cos(x)^4 

= 1 - 3cos(x)^2

Oh yay! My LaTeX finally worked for once. 

The definition of a rational number is any number that can be expressed as the quotient or fraction p/q of two integers.

For example, 1/7 = 0.142857142857...

This keeps going, it never ends but it is still rational. 

π, on the other hand is irrational because there is no pattern: 3.14159265358979323846...

Your number, as it appears has no pattern so it must be irrational.

Hey CPhill. Thanks for the answer, but you accidentally dropped the 2 on the cos⁴x.

All you need to do in this problem is plug in the numbers you know into the given function. 

2, 4, 16, 64, and you keep going...

You have $5$ red shirts, $5$ green shirts, $6$ pairs of pants, $8$ green hats and $8$ red hats, all of which are distinct. How many outfits can you make consisting of one shirt, one pair of pants, and one hat without having the same color of shirts and hats?

Options 

Red shirt, pants, green hat = 5*6*8 = 240 possibilities

or

Green shirt, pants, red hat = 5*6*8= 240 possibilities

Total = 240+240 = 480 possible combinations   

I don't see how you can avoid doing the integration again, since r is now a function of t.  The result I get when so doing is as follows:

(f + g)(x)   =   f(x) + g(x)             Since  f(x)  =  x³ - x ,  we can replace  f(x)  with  x³ - x .

(f + g)(x)   =   (x³ - x) + g(x)        Since  g(x)  =  x³ + 2x² - 10 ,  we can replace  g(x)  with  x³ + 2x² - 10 .

(f + g)(x)   =   (x³ - x) + (x³ + 2x² - 10)        We can drop the parenthesees.

(f + g)(x)   =   x³ - x + x³ + 2x² - 10         Addition can be done in any order.

(f + g)(x)   =   x³ + x³ + 2x² - x - 10         Combine like terms.

(f + g)(x)   =   2x³ + 2x² - x - 10

1.

(f - g)(x)   =   f(x) - g(x)

(f - g)(x)   =   (x⁴ - x² + 9) - (x³ + 3x² + 12)

(f - g)(x)   =   x⁴ - x² + 9 - x³ - 3x² - 12

(f - g)(x)   =   x⁴ - x³ - 4x² - 3

2.

(f ⋅ g)(x)   =   f(x) ⋅ g(x)

(f ⋅ g)(x)   =   (x⁴ - 9) ⋅ (x³ + 9)

(f ⋅ g)(x)   =   (x⁴ - 9)(x³ + 9)

(f ⋅ g)(x)   =   (x⁴)(x³) + (x⁴)(9) + (-9)(x³) + (-9)(9)

(f ⋅ g)(x)   =   x⁷ + 9x⁴ - 9x³ - 81

     x - 3

_____________

 x^2 + 7x  - 30

Notice that the denominator factors as

(  x  - 3 )                                          1

______________    =                 _____

 (x - 3) ( x + 10)                          x + 10

We will have a vertical asymptote at  x + 10  = 0  ⇒  x  = -10

And we will have a "hole"  at  x -3  = 0   ⇒  x  = 3

See the graph, here :  https://www.desmos.com/calculator/skml1smvwr

[The hole doesn't show....but. trust me...there is one at x  = 3   !!!  ]

    4x                                  A                   B

__________     =          ______   +       _____

x^2 - 8x + 15                  x  - 3               x - 5

This is known as a partial-fraction decomposition

Note that the denominator on the left factors as  ( x - 3)  (x  - 5)

Multiply both sides by this and we have

4x   =  A(x - 5)   +  B ( x - 3)     simplify

4x  = Ax - 5A + Bx - 3B

4x + 0 =  (A + B)x  + (- 5A -3B)

Equating coefficients, we have this system

A + B  = 4         ⇒  B  = 4 - A      (1)

-5A  - 3B  =  0   ⇒  5A  + 3B  =  0     (2)

Sub (1)  into (2)   and we have that

5A + 3(4 - A)  = 0

5A + 12 - 3A  = 0

2A  =  -12

A  = -6

And  B  =  4 - A   =   4 - (-6)   =  10

Verify  that

-6/ ( x - 3)  +  10/ (x  - 5)     =   the left hand side of the original equation

\(0\leq x \leq10 and 0\leq y \leq10\)

\(2x+5y \geq 20 \)

The first  two conditions set  up  a square that is  10 units  on a side ....so the total area  is  10^2  =

100 units^2  (1)

The area below  the  inequality   2x  + 5y  ≥ 20  [ but still within the square ]  forms a triangle  with a base  of 10 and a height of 4

Its  area  =  10  * 4 / 2  = 20 units^2

Then  the area  above the inequality  [ but still within the square is 100 - 20 ]  =  80 units^2   (2)

So....the probability that  the coordinates of a point satisfy all three conditions  is

(2) / (1)    =     80 / 100    =   4/5

Here's a graph : https://www.desmos.com/calculator/ome3jb01pa

The total number of pairs  =  C(8,2)  =  28

Adjacent pairs

(1,8) (1,2) (2, 3) (3,4) (4, 5) (5, 6) (6, 7) (7, 8)

So....the probability  is

8 / 28    =

2 / 7

sin^4x  - sin^2xcos^2x  -cos^4x

sin^4x  -  (1 - cos^2x)cos^2x - cos^4x

sin^4x - cos^2x + cos^4x - cos^4x

sin^4x  - cos^2x

Note :  other answers may be acceptable....!!!

The area  below l  and still in the first quadrant  is a triangle with a base of 6 and a height of 6

So...its area  is    base * height /2  =  6 * 6 /2  =   18 units^2   (1)

The area between the two lines also forms a triangle with a height of 6  and a base of 4.5

So...its area   is   6 * 4.5  / 2  =  13.5  units^2    (2)

So...the probability of a point being selected in the specified area  is   (2)  / (1)  =

13.5  / 18   = 0.75

Thank you very much I will think on this hard as suggested!

p.s. it's dom6547 not dom6527!