geno3141

For the children:  the pattern is 2, 3, 4, ...                           (adding 1 each time)

For the animals:  the pattern is 18, 22, 26, ...                      (adding 4 each time)

These are both arithmetic sequences because each increase by adding the same amount each time.

I will answer these by using formulas from the mathematical area of arithmetic sequences and series. If you are not familiar with these, they can be done without the formulas, but it will take longer. Post back if you want the more basic way.

The nth term of an arithmetic sequaen can be found by the formul:                                                                tn = t1 + (n-1)d         where:   tn is the nth term (or last term)      t1 is the first term                                                                                     n is the number of terms                d is the common difference

To solve this problem we are starting with the first new terms in the sequences (5 for children and 30 for animals)

For children, the first term will be 5, the number of terms is 52, and the common difference is 1.

Putting these into the equation, we get:  tn = 5 + 51(1) = 56

For animals, the first term will be 30, the number of terms is 52, and the common difference is 4.

Putting these into the equation, we get:  tn = 30 + 51(4) = 234

If I read the problem correctly, I will need to know the total number of tweets more for animals than for children; so if I find the total number of tweets for animals and subtract the total number of tweets for children, I will have the answer.

For arithmetic series, the sum has this formula:  S = n(f + l) / 2                                                                 where S is the sum     n is the number of terms     f is the first term     and     l is the last term.

For children, n = 52, f = 5, and l = 56.

Putting these into the equation, we get:  S = 52(5 + 56) / 2  =  1716  total tweets for children.

For animals, n = 52, f =30, and l = 234.

Putting these into the equation, we get:  S = 52(30 + 234)/2  =  6864 total tweets for animals.

The difference is 6864 - 1716 = 5148.

Let n be the number.

Left side of the equation:

The product of the number and -3 becomes -3n

Reducing this by 5 means subtracting 5 from this:  -3n - 5

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Right side of the equation:

The opposite of the number is -n

Twice this is -2n

25 less than this is  -2n - 25

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Equation:                                    -3n - 5  =  -2n - 25

Adding 3n to both sides:      -3n + 3n - 5  =  -2n + 3n - 25

                                                        -5  =  n - 25

Add 25:                                             20  =  n

Was this the question:  What was the most likely score?  It would be the average score.

A very small number.

(.3)^264  =  (3/10)^264  =  (3^264) / (10^264)

My calculator won't calculate (.3)^264 and it won't calculate (.3)^264 , so I'll need to use a smaller number.

You can use the calculator to get 3^132, which is approximately 10^63.   (Check this out.)

3^264 ≈ (3^132)^2             (When you have an exponent to an exponent, you multiply the exponents)

           ≈ (10^63)^2 

           ≈ 10^126

(3^264) / (10^264) ≈ (10^126) / (10^264)  ≈  10^ -138

Sincd the 3rd equation has only x's and z's, combine the first two equations to get rid of y's:

  -2x + y - z = 2           ---> multiply this equation by 3 --->  -6x + 3y - 3z = 6

   -x -3y + z = -10        ---> leave this equation alone   --->     -x - 3y  + z = -10

                                                                                         ----------------------

                    add down the columns:                                    -7x         -2z = -4

Combine this new equation with the original 3rd equation:

    -7x - 2z = -4            ---> multiply this equation by 3 --->  -21x - 6z = -12

    3x + 6z = -24           ---> leave this equation alone   --->     3x + 6z = -24

                                                                                           ------------------

                      add down the columns:                                    -18x        = -36    -->  x  = 2

Now, replace x with 2 in 3x + 6z = -24, to get the value for z,

and then replace those values into one of the first two original equations to get the value for y.

If the fraction is like  3/4, divide 4 into 3.

If the fraction is like 2 5/6, turn the fraction into a common fraction (2 x 6 + 5 = 17) = 17/5, then divide 5 into 17.

Rearrange as:  7d - 4d + 12 - 3  =  (7d - 4d) + (12 - 3)

Combine 7d - 4d to get 3d

Combine 12 - 3 to get 9

=  3d + 9

Could that be the subject of your PhD dissertation in mathematics?

You need to know the other leg, or that the triangle is isosceles, or the size of one of the acute angles ...

Part b:  y = (1/3)x^3 + (1/2)x^2 + (1/4)x

            y'  =  3(1/3)x^2 + 2(1/2)x^1 + 1(1/4)x^0

            y'  =  x^2 + x + 1/4             The derivative of a constant is zero.

            y''  = 2x + 1

c)  s = -2t^-1 + 4t^-2                    (Write as negative exponents; it's easier.)

     x' = -1(-2t^-2) + -2(4t^-3)

     x' = 2t^-2 - 8t^-3

     x'' = -4t^-3 + 24t^-4

d)  r = 12θ^-1 - 4θ^-3 + θ^-4

     r' = -12θ^-2 + 12θ^-4 - 4θ^-5

     r'' = 24θ^-3 - 48θ^-5 + 20θ^-6 

e)  r  = 2θ^(-1/2) + 2θ^(1/2)              Easier to work if negative, fractional exponents.

     r'  = -θ^(-3/2) + θ^(-1/2)               Subtract 1 from each exponent

     r'' = (3/2)θ^(-5/2) - (1/2)θ^(-3/2)

f)  w = re^(-r)

     w' = -re^(-r)                           (Do you have exponentials?)

g)  y = x^(-3/5) + π^(3/2)            π^(3/2) is a constant, so its derivative is zero

     y' = (-3/5)x^(-8/5)                  Subtract 1 from the exponent