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The answer is 4p^5*q^8

What is the value of y in the arithmetic sequence y + 6, 12, 3y?   

Set it up with the interval in an equality.       (12) – (y + 6)  =  (3y) – (12) 

                                                                         12 – y – 6  =  3y – 12 

Let's put all the "Y"s on one side and  

all the numbers on the other side                                 18  =  4y  

Swap sides because I like the unknowns 

to be on the left.  No other reason.                                4y  =  18  

Divide both sides by 4                                                     y  =  18/4 

                                                                                        y  =  4.5  

Check answer:   (y + 6) , 12 , 3y  

                          (4.5 + 6)  ,  12  ,  (3 • 4.5)  

                           10.5  ,  [add 1.5]  12  ,  [add 1.5] 13.5  

                           looks good  

This is trivial if you use coordinates.

Your answer is not useful.

It is not a trick but pretty much any scientific calculator can do it.

You have a square.  All the sides are the same. the perimeter is 20 what is the side length?

Then what is the area of this square?

You have a another square.  All the sides are the same. the perimeter is 8 what is the side length?

Then what is the area of this square?

Hi Guest!

This is the question:\(\text{The function }f(\sqrt{x+1})=\dfrac{1}{x+1} \text{ satisfies for all } x \ge -1, x\neq 0. \text{ Find f(2)}\)

To find \(f(2)\) we need to make whatever inside the bracket of "f" to be 2.

I.e. 

\(\sqrt{x+1}=2\\ \iff x+1=4 \implies x=3\)

So, if we substitute x=3 in the function we get the answer.

That is, \(f(\sqrt{3+1})=f(\sqrt{4})=f(2)=\dfrac{1}{3+1}=\dfrac{1}{4}\)which is the desired answer.

I hope this helps, and don't hesitate to ask for further clarificiation or any other question :) .

The sequence \( a_n\) has the property that \( a_n = a_{n-1} + 2a_{n-2}\) for \( n \ge 2\). It is also true that \(a_0 = 4\) and \( a_3 = 6\). What is the value of \(a_5\)? Express your answer as a common fraction.

For n = 3 and n = 2:

\(a_3 = a_2 + 2 a_1\\ a_2 = a_1 + 2a_0\)

substituting,

\(6=a_2+2a_1\\ a_2 = a_1+8\\ \)

this is a simple system of equations.

we can find \( a_n\) for any n now by plugging \(a_{n-1}\) and \(a_{n-2}\) into the formula.

For example, \( a_4 = a_3 + 2a_2\) (which we know already)

so you would split 20 and 8 into fourths?  I am confused

Here is an earier solution to this question by CPhill

You are very welcome.

It is not a proof though...

0.17  x  10,250 =~1,742 families with 1 child [Not rounded]

0.43  x  10,250 =~4,407 foamilies with 2 children [Not rounded]

0.28  x  10,250 =2,870 families with 3 children

0.12  x  10,250 =1,230 families with 4+ children

10,250  -  [1,742 + 4,407 + 2,870 + 1,230] = 1 family has no children !. 

What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?  

Let's rearrange 6x - x^2 = k to standard quadratic equation form.   –x² + 6x – k = 0  

Multiply both sides by –1 to get that pesky negative off the x²          x² – 6x + k = 0  

What two numbers can we add together to get 6?  

There are 5 + 1,  4 + 2, and 3 + 3.  

What do we get when we multiply each of those pairs together?  

There are 5•1=5, 4•2=8, and 3•3=9.  

9 is the largest, so let's say that k = 9.  Now we have                        x² – 6x + 9 = 0  

This will factor to                                                                                (x – 3)(x – 3)  

GCD {[121^2 + 233^2 + 345^2] ,  [120^2 + 237^2 + 348^2]}

Prime factorizations

187955 = 5×37591 (2 distinct prime factors)
191673 = 3^3×31×229 (5 prime factors, 3 distinct)

Therefore: GCD = 1

3^6 =729 ways - without restrictions. 

When you look at the Top Users list on the right, CPhill seems to be not there ... at first, that is. 

If you click the down arrow at the bottom of the Top Users field, Chris's name appears at the top. 

_.

What is the greatest common divisor of 121^2 + 233^2 + 345^2 and 120^2 + 237^2 + 348^2?  

Just checking if the first "Guest" answer is even plausible. 

    121²    +    233²    +    345²  

= 14,641  +  54,289  +  119,025  

=  187,955    This is not evenly divisible by 34 so the first "Guest" is a liar. 

The 5th power of a positive number is equal to the product of 1/3 and the 6th power of the number. What is the ratio of this number's 12th power to its 9th power? Express your answer as a common fraction.  

  x⁵        x⁶  

–––  =  –––      Dividing both sides by x⁵,    x = 3  

  1         3  

  x¹²          x³  

––––   =  ––––  =   3³  =  27  

   x⁹           1  

How did you even get 1/2? and how does calculus come in play bruh?

0 < n < 100

\(5^n + 8^{n + 1} + 11^{n + 2} + 5^{n+2} \)

and the expression above is divisible by 6.

5^n will always end in 5.

11^m will always end in 1.

That means 5^n + 5^(n + 2) will end in 0.

8^(n + 1) will end in an even number, meaning that the expression will be odd because of 11^(n+2). 

If a number has to be even to be divisible by 6, then that means:

The expression will never be divisible by 6 for 0 < n < 100.

Exponent property \((a^m)^n = a^{mn}\)

So (a^9)^5 = a^(9 * 5) = a^45

When you divide by a fraction, you multiply by the reciprocoal, which is 1/(10/11) = 11/10 (flip numerator with denominator).

Thus, 5/9 divided by 10/11 is 5/9 times 11/10 = 55/90. Divide top and bottom by 5 and you get 11/18.

Thanks melody, I liked your reasoning.

Yes the answer was k = 1 for the optimal solution --- thanks, proyaop.

The ratio is 3/5.

The gcd is 34.