geno3141

First, rewrite  ( 3√z - 1 ) / ( 1 - √z ) by multiplying both the numerator and denominator by -1:

                 =  ( 1 - 3√z ) / ( √z - 1 )

Now, you have:

   ( 2√z - 3 ) / ( √z - 1 )  -  1   =   ( 1 - 3√z ) / ( √z - 1 )

Multiply each term by (1 - √z ) / ( √z - 1 ) - 1

     2√z - 3  -  1 · ( √z - 1 )  =  1 - 3√z 

     2√z - 3 - √z + 1  =  1 - 3√z

                   √z - 2  =  1 - 3√z

                        4√z  =  3

                          √z  =  3/4

                            z  =  9/16

8a² - 72  =  0

Add 72 to both sides:

8a²  =  72

Divide both sides by 8:

a²  =  9

Find the square root of both sides;

a  =  3     or     a  =  -3

A right triangle is formed with one corner at Dan, another corner, the right angle, at the building (6 feet from the base), and the other corner being the top of the building.

The distance from Dan to the building (6 feet from the bottom) is the adjacent side = 30.

The distance from the building (6 feet from the bottom) to the top of the building is the opposite side = 59.

tan (angle) = opposite / adjacent     --->     tan(angle)  =  59/30

--->     angle  =  tan^-1(59/30)

--->     angle  =  63°

A geometric series converges only when the common ratio is a number strictly between -1 and +1.

Since the common ratio (the number that you use to multiply a term to get the next term) is 3, this series diverges.

log₂(x + 1) + log₂(x + 7)  =  3

log₂[ (x + 1)(x + 7) ]  =  3

Simplifying the left side:

log₂[ x² + 8x + 7 ]  =  3

Write in exponential form:

x² + 8x + 7  =  2³

x² + 8x + 7  =  8

x² + 8x - 1  =  0

Now, finish by using the quadratic formula (You will need to reject imaginary answers and any answer that is less than or equal to 0 (because that will create a log of a non-positive number, which is undefined).)

If the sequence is B, C, E, H

and if you replace the letter with its place in the alphabet,

then the sequence becomes:  2, 3, 5, 8.

This could be a Fibonacci sequence where, after the first two terms, the next term is found be adding the preceding two terms. 

This results in the next term being 5 + 8  =  13, which corresponds to letter M.

The negative signs are bothersome, but do not invalidate the result ...

If you want to solve this problem using formulas, there are two important formulas:

t₁  =  first term  =  -5          

t_n  =  n^th term  =  last term  =  -5

n  =  numbr of terms

S_n  =  sum of the n terms

t_n  =  t₁ + (n - 1)d     --->     65  =  -5 + (n - 1)(5)

                                          65  =  -5 + 5n - 5

                                          65  =  -10 + 5n

                                          75  =  5n

                                            n  =  15

S_n  =  n(t₁ + t_n)/2     --->     S_n  =  15(-5 + 65)/2

                                          S_n  =  450

Any 5-digit multiply of 12 must be at least 10000.

Because 12 is 4 x 3, any multiply of 12 must be divisible by both 3 and 4.

To be a multiply of 3, the sum of the digits must be divisible by 3.

Since the only digits are either 1 or 0, the number must contain three 1s.

To be divisible by 4, the number must be even, so it must end with a zero; and, since its "twice-even", it must end with two 0s.

11100 fills the requirements...

1! = 1     2! = 2     3! = 6     4! = 24     5! = 120     6! = 720     7! = too many digits

Adding  1!  to  2!  to  5!  results in 125.

Being divisible by 1 doesn't help, because all whole numbers are divisible by 1.

Being divisible by 2 means that the number  n+ 2  is even, so n must also be even.

Being divisible by 3 means that, if the number  n + 3  is divisble by 3, so  n  must also be divisible by 3.

Being divisble by 4 means that, if the number  n + 4  is divisble by 4, so  n  must also be divisible by 4 (and this number must be even).

Being divisble by 5 means that, if the number  n + 5  is divisible by 5, so  n  must also be divisible by 5.

The smallest number divisible by 3, 4, and 5 is  3x4x5  =  60.