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I have a more direct solution than you. 

a = a (vertically opposite angle) <-- This means the angle in the triangle is also 'a degrees'

a + b = 107 (exterior angle of triangle)

Finished!! :)

And n(n+1)/2 = 1 + 2 + ...... + n can be proved by mathematical induction.

Let P(n) be n(n+1)/2 = 1 + 2 + ....... + n

For n = 1,

LHS = 1(1+1)/2 = 1

RHS = 1 = LHS

P(1) is true.

Assume that P(k) is true,

For n = k + 1

 LHS 

= (k+1)(k+2)/2

 RHS 

= 1 + 2 + ...... + k + (k + 1) 

= (k + 1)(k)/2 + (k + 1)

= \((\dfrac{k+1}{2})(k + 2)\)

= (k+1)(k+2)/2

= LHS

Therefore when P(k) is true, P(k + 1) is true.

By the definition of mathematical induction, P(n) is true for all positive integers n.

The wrong step is at the step n*(n+1) = 2n^2.

n * (n + 1) = n^2 + n instead.

\(n(n+1) = n^2 + n\)

\(n^2 + n = 20\\ n^2 + n - 20 = 0\\ (n+5)(n-4) = 0\\ n = -5(\text{rejected})\text{ or }n=4\)

Now that's correct.

Let a = arctan(\(\sqrt2 - 1\))

Double angle formulae:

\(\tan(2a)\\=\dfrac{2\tan a}{1-\tan ^2 a}\\=\dfrac{2(\sqrt2 - 1)}{1-(\sqrt2 - 1)^2}\\ = \dfrac{2\sqrt2 - 2}{1-(2+1-2\sqrt2)}\\ =\dfrac{2\sqrt2 - 2}{1-2-1+2\sqrt2}\\ =\dfrac{2\sqrt2 - 2}{2\sqrt2 - 2}\\ =1\)

When tan(2a) = 1, 2a = 45 degrees. Therefore a = 22.5 degrees or pi/8 radians.

8(arctan(sqrt(2)-1)) = 8(pi/8) = pi.

Using Law of Cosines, we have the following results:

Sides:
a = 4 m
b = 5 m
c = 6 m

Angles:
A = 41.4096°
B = 55.7711°
C = 82.8192°  Therefore: C=2A. The naive Proof!!.

\(4xy^z+4=6\\ 4xy^z=2\\ xy^z=\dfrac{1}{2}\\ x=\dfrac{1}{2y^z}\\ y^z=\dfrac{1}{2x}\\ y = \dfrac{1}{\sqrt[z]{2x}}\\ z\log y = -\log (2x)\\ z = -\dfrac{\log2x}{\log y}\)

En karta ritade i skal i 1:250 000. Avståndet mellan två byar på kartan är 6 cm.

Hur många kilometer är det verklighetn?

=?

to work out m i need an answer to the equation

After t hours A will have walked 5t km north.

After t hours B will have cycled 16(t-1) km east.

The distance between them is given by Pythagoras:   sqrt( 5²t² + 16²(t-1)² ) → sqrt(281t² - 512t + 256) km

So we have:  sqrt(281t² - 512t + 256) = 52

Square both sides 

281t² - 512t + 256 = 2704

281t² - 512t - 2448 = 0

(281t + 612)(t - 4) = 0

The only positive solution is t = 4 hours.

Hence they are 52 km apart at 7 + 4 → 11 am.

Hmm.

.

Guest #2 has the right idea, but has made a mistake in his cross-multiply stage.

.

This is just (999999 - 999991)^2 = 8^2 → 64

.

Note that (a - b)^2 = a^2 + b^2 - 2*a*b

.

It equals sqrt(2).

Guest #2:

This is a simple "continued fraction" and has nothing to do with "x" and the "quadratic formula"!!!!!!!; It  simply = 8 /11.

Let's call this fraction x. Now forget about the\(\frac{1}{1+\frac{1}{2+}}\)

Now the rest is exactly the same as x.

Thus, \(\frac{1}{1+\frac{1}{2+x}}=x\)

Doing cross multiplication, we get \(1=x+\frac{x}{2+x}=\frac{x^2+x}{x+2}\)

Doing cross multiplication again(on the left and right side), we get: \(x+2=x^2+x \rightarrow 0=(+1)x^2+(+0)x+(-2).\)

In this case, a=+1, b=0, c=-2. Using the quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

We get x=\(\frac{+\sqrt{8}}{2}\)or \(\frac{-\sqrt{8}}{2}\)

Since all of 1,1,2,1,1,1,2,1,1,1,2,1.... are all positive, x must be positive.

So, x=\(\frac{\sqrt{8}}{2}=\sqrt{2}\)(approximation: 1.41421356...)

BIG BUG>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>..............................................

107 degree + ? = 180 degree (adjacent angles on stright line )

  ? = 180-107 

     = 73 degree 

the angle next to 107 degree is 73 degree 

since the angle next to 107 degree is in the triangle ,

we can use angle sum of triangle to solve this problem

 the angle oppsite to angle A degree is  

 a=a ( vertical oppsite angle )

therefore the angle oppsite to a degree is also a degree

after that we finall need to find the sum of a + b

you should know that the sum of the angle degree in a triangle is 180 degree , right ?

threrefore 

 a degree + b degree + 73 degree = 180 degree (angle sum of triangle)

                                                  a+ b= 180-73

                                                        =107 degree

therefore the sum of a degree + b degree is 107 degree

Due to the fact that 1248.00=1248 and 0.2=1/5, \(1248.00*0.2=1248*\frac{1}{5}=\frac{1248}{1}*\frac{1}{5}=\frac{1248}{5}=249.6\)

Or it could be: \((2*5)-\sqrt{9}=7\)

SEE THE ANSWER IN THE MIDDLE OF THE PAGE POSTED UNDER THE NAME "higgsb".

[9 + 5] / 2 = 7

Sorry, I forgot to mention that you can always enter the the left  numbers [0; 1,2,1,2] as "continued fraction" in W/A engine like this:                                                                                                   http://www.wolframalpha.com/input/?i=continued+fraction%5B0;1,2,1,2%5D.

And will always give the fraction.

4xy^z+4 =6

4xy^z = 2

xy^z = 1/2 

Threre are INFINITE solutions to this...need more info....

Since ABCDE is a regular pentagon, all sides are congruent.

Specifically side(AB) = side(BC).

Therefore,in triangle ABC, angle(BAC) = angle(BCA).

Since the sum of the angles of a triangle is 180°, and angle(ABC) = 108°, leaving 72° for the sum of angle(BAC) and angle(BCA).

Therefore, angle(BAC) is one-half of 72°, or 36°.