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22/15

thankyou your. the best. i actually calculated the answer on some random website and it took me a while but i got 7.69% as an interest rate but the thing is on the website it didn't have any working out or formula's it just gave me the answer. i klnew it was right. and i think all you needed to do was multiply your final answer by 2 and i got the answer. so i'm very much appreciative of your work. you've helped me so much. i hope good things come to you... again thankyou. your a legend

I assume that you mean complex number rather than complete number.

There are two obvious methods, first, you can build upto the result by a series of multiplications and squarings.

So for starters (z+1/z)^2 = z^2 + 2 + 1/z^2 = 1,  so  z^2 + 1/z^2 = -1.

Squaring that, z^4 + 2 + 1/z^4 = 1, so z^4 + 1/z^4 = -1 and so on.

The intermediate expression z^3 + 1/z^3 can be obtained by multiplying z + 1/z by z^2 + 1/z^2 etc..

A second method requires you to solve the equation  z + 1/z = 1.

That gets you z = 1/2 +- i*sqrt(3)/2 = cos(pi/3) +- i*sin(pi/3) and now you can use De'Moivre's theorem.

(a) Compute the sums of the squares of Rows 1-4 of Pascal's Triangle. That is, compute:

\(\begin{array}{|lcll|} \hline \binom10^2 + \binom11^2 &=& \binom21 =2 \\ \binom20^2 + \binom21^2 + \binom22^2 &=& \binom42 = 6 \\ \binom30^2 + \binom31^2 + \binom32^2 + \binom33^2 &=& \binom63 = 20 \\ \binom40^2 + \binom41^2 + \binom42^2 + \binom43^2 + \binom44^2 &=& \binom84 = 70\\ \hline \end{array}\)

Do these sums appear anywhere else in Pascal's Triangle?
Yes: as Central binomial coefficients

(b) Guess at an identity based on your observations from part (a). Your identity should be of the form

\(\begin{array}{|rcll|} \hline \binom{n}{0}^2 + \binom{n}{1}^2 + \cdots + \binom{n}{n}^2 &=& \binom{2n}{n} \\ \hline \end{array}\)

Okay, so what we're looking at here is exponentials and logarithms.

note; my first assumption is that Semiannual means "one half year" or "two times a year", so in question 1. we're really looking at 8*2 = 16 payouts.

For question 1 (as an example):
firstly, normalise the repay so you know how much (as a factor) you've gained:
\( {64010 \over 35000} = 1.82885714....\)

This makes our maths a lot easier.

Next thing we want to know, is what percentage increase (interest rate) do we want to compound every 6 months, so that after 8 years (or 16 iterations) we get to 1.82885714...

The "compound" part basically means that every time you gain interest, next time you'll be getting interest on your new balance:
5% interest, 3 years: 1 * 1.05 = 1.05 (one year) , 1.05 * 1.05 = 1.1025 (second year), 1.1025 * 1.05 = 1.157625 (third year) etc...

*Right, nearly at the end now!*

In the 5% interest example above, we knew what percentage we had, and how long we are compounding the interest for, and we found the result after 3 years.

However, for question 1, we don't know the interest, but we do know the end result = 1.82885714.

Your ultimate formula is

\(( 1 + interest ) ^ {iterations} =end value\)

converted to:

\({endvalue} ^ {1 \over {iterations} } = (1 + interest)\)

(I'm assuming you've met "powers" before, i.e. 2^2 = 4, 2 ^ 3 = 8, 3 ^ 5 = 243 etc).

so your interest for Q1 is:

\(1.828857^{1 \over 16} = 1.03845\)

interest = 3.845%

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"Ts= 2(pi) sqrt((4.5*10-2kg)/(2.0*103kg/s2))"

\(\frac{4.5\times10^{-2}kg}{2.0\times10^3kgs^{-2}}\rightarrow 2.25\times10^{-5}s^2\)

So:  \(Ts=2\pi\sqrt{2.25\times10^{-5}s^2}\rightarrow 0.03s\)

.

Just put it in exactly as you would on paper.

You can only use round brackets though and don't leave any spaces.

*   means muliply and     / means divide  and   ^ means to the power of.

"During my trip to work, I spent 3/4 of the time driving on the highway, during which I covered 5/6 of the distance. My average speed on the highway was 60 miles per hour. What are the possible values of my average speed, in miles per hour, during the time I was not on the highway?"

.

(a/b)^m = a^m/b^m - TRUE if a, b and m are positive.

(a^m)^n = a^m(n) - TRUE if a, m and n are positive.

(a/b)^ m(n) =(a^m/b^m)^n - TRUE if a, b, m and n are positive.

Verify the following identity:
sin(x)+(sin(3 x))/(3) = 2 sin(x)-(4 sin(x)^3)/(3)

Put sin(x)+1/3 sin(3 x) over the common denominator 3: sin(x)+1/3 sin(3 x) = (3 sin(x)+sin(3 x))/3:
(3 sin(x)+sin(3 x))/3 = ^?2 sin(x)-(4 sin(x)^3)/(3)

Put 2 sin(x)-(4 sin(x)^3)/3 over the common denominator 3: 2 sin(x)-(4 sin(x)^3)/3 = (6 sin(x)-4 sin(x)^3)/3:
(3 sin(x)+sin(3 x))/3 = ^?(6 sin(x)-4 sin(x)^3)/3

Multiply both sides by 3:
3 sin(x)+sin(3 x) = ^?6 sin(x)-4 sin(x)^3

sin(x)^3 = 1/4 (3 sin(x)-sin(3 x)):
3 sin(x)+sin(3 x) = ^?6 sin(x)-4(3 sin(x)-sin(3 x))/(4)

-(3 sin(x)-sin(3 x)) = sin(3 x)-3 sin(x):
3 sin(x)+sin(3 x) = ^?6 sin(x)+sin(3 x)-3 sin(x)

6 sin(x)-3 sin(x)+sin(3 x) = 3 sin(x)+sin(3 x):
3 sin(x)+sin(3 x) = ^?3 sin(x)+sin(3 x)

The left hand side and right hand side are identical:
Answer: |(identity has been verified)

Sine(x) =0.045

x =Sine^ -1(0.045)

x=2.5792 degrees - the angle.

Daaaamn never thought about the log base e (e)= 1 thing. So obvious.

Thanks a huge lot CPhill, you're helping me a lot. 

Is this correct?

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If the length is x miles and if the time was y mins... {nl} 5x/6 distance. {nl} 3y/4 time.

Let's say that it was 60 miles, and the drive took an hour. {nl} Then, for 15 mins, they covered 10 miles. {nl} Times 4 for 15 mins and 10 miles. {nl} Thus, the average speed is 40mph. {nl}  

2^27+3^27?=134,217,728 + 7,625,597,484,987 =7,625,731,702,715

Prime factors of:7,625,731,702,715 =5 * 7 * 577 * 377604937

Therefore: 5 + 7 =12 sum of 2 smallest prime factors.

log_b k  = 2.5       means that  b^2.5  = k

log_e b = 2         means that e^2  = b

So  this means that   b^2.5  = k  →   (e^2)^2.5  = k   →   e^5  = k

So 

log_e k   = log_e e^5   =    5 log_e e  =  5 * 1    =  5

1/2(x-3)=5/12 + 2/3(2x-5)        multiply through by 12

6(x -3)  = 5 + 8(2x - 5)     simplify

6x - 18   = 5 + 16x - 40

6x - 18 =  -35 + 16x      add 35 to both sides, subtract 6x from both sides

17  = 10x       divide both sides by 10

17 / 10   = x

1/x +  1/y  = 1/7   simplify

[x + y ] / xy  =  1/7

7[x + y]   = xy

7x + 7y - xy   = 0          rewrite as

xy - 7x - 7y   = 0        add 49 to both sides

xy - 7x - 7y +  49  =  49      factor

(x - 7) ( y - 7)   = 49       (1)

Note that the following  ordered pairs of integers would work :

(14, 14)

(8, 56)

(56, 8)

(-42, 6)

(6, -42)

Note that (0,0)   would  also satisfy (1), but  ......  x and y cannot be 0 in the original problem  [division by 0 ]

The s2 is actually suppose to be s^2

Here is a more "mathematical"  way of doing this......

2^27   +  3^27   =

[ (2⁹)³ + (3⁹)³ ]  =  

[ 2⁹ + 3⁹ ] [ 2¹⁸  - (2 * 3)⁹  + 3¹⁸ ]  =

[ (2³)³ + (3³)³ ]  [ 2¹⁸  - (2*3)⁹  + 3¹⁸ ]  =

[ 2³ + 3³] [ 2⁶ - (2 * 3)³ + 3⁶] [ 2¹⁸  - (2*3)⁹  + 3¹⁸ ]  =

[2 + 3] [ 2² - 6 + 3² ]   [ 2⁶ - (2 * 3)³ + 3⁶] [ 2¹⁸  - (2*3)⁹  + 3¹⁸ ]  =

[5] [ 13 - 6] [577] [377604937]  =

[ 5 ] [ 7 ]  [577] [377604937]  

So.......the sum of the two smallest distinct prime factors of  2^27 + 3^27    =    12

Lol can I delete a comment. I read the question wrong

Well think about it. 6 tenths is 6/10 or 3/5. So if an hour has 60 minutes, multiply 3/5 times 60 and you get 36 minutes. Hope this helps

An hour has 60 minutes.

So, 1/6*60=10 min.