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 For how many integers n between 1 and 11 (inclusive) n/12 is a repeating decimal?

\(\begin{array}{|r|r|r|r|r|} \hline & & & & & \text{prime number only } \\ & & & \text{factorize} & \text{prime numbers} & 2 \text{ or } 5 \\ n & \dfrac{n}{12} & \text{cancel} & \text{the denominator} & \text{in the denominator}& \text{will terminate} \\ \hline 1 & \dfrac{1}{12} & \dfrac{1}{12} & \dfrac{1}{2^2\cdot 3} & 2,\ 3 \\ \hline 2 & \dfrac{2}{12} & \dfrac{1}{6} & \dfrac{1}{2 \cdot 3} & 2,\ 3 \\ \hline 3 & \dfrac{3}{12} & \dfrac{1}{4} & \dfrac{1}{2^2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 4 & \dfrac{4}{12} & \dfrac{1}{3} & \dfrac{1}{3} & 3 \\ \hline 5 & \dfrac{5}{12} & \dfrac{5}{12} & \dfrac{1}{2^2\cdot 3} & 2,\ 3 \\ \hline 6 & \dfrac{6}{12} & \dfrac{1}{2} & \dfrac{1}{2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 7 & \dfrac{7}{12} & \dfrac{7}{12} & \dfrac{7}{2^2\cdot 3} & 2,\ 3 \\ \hline 8 & \dfrac{8}{12} & \dfrac{2}{3} & \dfrac{2}{3} & 3 \\ \hline 9 & \dfrac{9}{12} & \dfrac{3}{4} & \dfrac{3}{2^2} & 2 & \checkmark \Rightarrow \text{terminate} \\ \hline 10 & \dfrac{10}{12} & \dfrac{5}{6} & \dfrac{5}{2\cdot 3} & 2,\ 3 \\ \hline 11 & \dfrac{11}{12} & \dfrac{11}{12} & \dfrac{11}{2^2\cdot 3} & 2,\ 3 \\ \hline \end{array} \)

Hi Guest,

\(\begin{array}{|rcll|} \hline && \mathbf{\cos(195^\circ)} \\ &=& \cos(180^\circ +15^\circ)\\ &=& \underbrace{\cos(180^\circ)}_{=-1}\cos(15^\circ) - \underbrace{\sin(180^\circ)}_{=0}\sin(15^\circ)\\ &=& -\cos(15^\circ) - 0\cdot \sin(15^\circ)\\ &\mathbf{=}& \mathbf{-\cos(15^\circ)} \\\\ &=& -\cos(45^\circ - 30^\circ) \\ &=& -\Big( \underbrace{\cos(45^\circ)}_{=\dfrac{\sqrt{2}}{2}} \underbrace{\cos(30^\circ) }_{=\dfrac{\sqrt{3}}{2}} + \underbrace{\sin(45^\circ)}_{=\dfrac{\sqrt{2}}{2}} \underbrace{\sin(30^\circ)}_{=\dfrac{1}{2}}\Big) \\ &=& -\dfrac14 *\sqrt{2}*(\sqrt{3}+1) \\ \hline \end{array}\)

m^-9   is    1 / (m^9)   

so  m^5 * m^-9  =  m^(5 + (-9))  = m^(-4)  =   1/m^4

Thank you CPhill,

Sorry GA,

I corrected my typo:

\(\boxed{\dfrac{1}{k(k+1)} = \dfrac{1}{k} - \dfrac{1}{k+1} } \)

No....I think your APPROACH was OK....the execution of it was wrong

remember cos pi/2 = 0     sooooo    sin (x+ pi/2) = sin x cos pi/2  + cos x sin pi/2       cos pi/2 = 0   and  sin pi/2 = 1

                                                         sin ( x + pi/2) =  0  + cos x       SEE?

Using  sin (A + B)  = sinAcosB + sin B cosA  

[ cos theta = -12/13]

So

sin ( theta + pi/2 ) =   

sin (theta) * cos(pi/2) + sin (pi/2) * cos (theta) =

 (5/13) (0)  + (1) (-12/13) =

-12/13

So was my approach, using the sum and difference identity, wrong?

THX,  Guest....!!!

That's an excellent answer  !!!!

Sin (x + pi) =   cos x         = - 12/13    (remember ...it is in quadrant II )

This uses the log property that says

a  ^{log _a b}     =   b

So 

7  ^{log ₇ x}   =   x       

Mmmmm....that's kind of a lot of stuff....

First one     ...    A) f(x) = 51.89x + 1137.5

Second ....   C) f(x) = 0.058x + 1.88; during the year 2020

Third  .....  C)  y = -292x + 2221; r = -0.98; predicted value at age 9  =  -292(9) + 2221  = -$407

Fourth...   B)  y = 2.8x + 15.1; r = 0.92;  2.8(8) + 15.1  = 37.5   (in 1000s)

Last One ... The approximate equation is

y = 12.7236x + 262.9

The budget will be   12.7 (11) + 262.9   = 402. 5

"A" seems closest

Look at log base 10      log 1000 = 3       10^(log1000) = 10^3 = 1000

now put in logbase 7    Log7  x =  y  

                                          which means   7^y  = x

                                                                       then  7^log7 x  = 7^y    which would equal  x

FIrst the large hemicircle of the 10 ft rope area = 1/2 pi r^2 =  50 pi

  when the pooch gets to the edge of the building he has a quarter circle (on BOTH sides of the building) of radius 7

      this area = 1/2 pi r^2 = 24.5 pi

total area = 74.5 pi   sq units

If a rational number's decimal representation doesn't terminate then it repeats.

A rational numbers decimal representation will terminate only if the denominator's only prime factors are 2 and 5.

With 12 in the denominator we need a 3 in the numerator to ensure the decimal terminates.

Thus 3/12, 6/12, 9/12 will all terminate

1/12, 2/12, 4/12, 5/12, 7/12, 8/12, 10/12, 11/12 will all repeat

Sorry Guest