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2* 3 * 5 * 7 *19   =  3990

2^2 * 3 * 5 * 7 * 19  = 7980

g(2x)  =  (2x + 3) / 4

2 ( g(x))  =  (x + 3) / 2

So

(2x + 3) / 4  =   (x + 3)  / 2      cross-multiply

2 (2x + 3)  = 4 ( x + 3)

4x + 6  = 4x + 12

No solution possible

Did you get the right answer from your prof?

Anchor Jill in any chair

The two boys can sit on either side of her in 2 ways

And the other two girls can be arranged in 2! = 2 ways

So   2 * 2  =   4 ways

Wow that is so cool! I know a lotttt of math sites, but never heard of this. Thanks for sharing!

Very nice, geno.....!!!

I believe that one answer is  x = 1  y = 4  z = 5.

Another answer is  x = 1  y = 3  z = 8.

sqrt (3) / 2  =   slant height  / 12

6sqrt (3) =  sqrt (108) = slant height

Height    = sqrt (108 - 36)  =  sqrt (72) = 6sqrt (2)

Volume   = 

(1/3) 12^2 * 6 sqrt (2)  = 

288 sqrt (2) units^3

*** EDITED ***

19 - 10 = 9

10 - 1 = 9

the 27th term is 1 + 9 * (27 - 1)

which is 1 + 9*26                                   (as first Guest posted)

which is 235

the value of the 27th term is 235    

f is said to be incresaing whenever the gradient of the tangent is positive.  When is that?

Find the height of the pyramid     to be sqrt 108         (use pythag theorem)

volume = 1/3 (12)(12) (sqrt(108))    units³

19 - 10 = 9

10 - 1 = 9

the 27th term is 1 + 9 * (27 - 1)

which is 1 + 9*28

which is 253

the value of the 27th term is 253

Another answer is 4 rows of 82 chairs.

*** EDITED ***

The other answer is for a GEOMETRIC sequence

Here is answer for ARITHMETIC sequence

find the common difference     10 - - 8  = 18

x2 + ( +18)(14) = - 8

x2 = -260

Notice the second term is 9 more than the first....

   the 27th term will be  26 * 9    bigger than the first

       1 +  26*9 = a₂₇  = ___________

$3,000  x  0.08 / 12 ==$20 - interest on the first month's payment.

2.5  x  1/6 ==5 / 12 of 1 pound or 6 2/3 ounces.

There are:

(9, 8, 7, 6, 5) , (8, 7, 6, 5, 4) , (7, 6, 5, 4, 3) , (6, 5, 4, 3, 2) , (5, 4, 3, 2, 1) , (4, 3, 2, 1, 0) , >Total = 6 combinations

(4, 3, 2, 1, 0) ==1 permutation

(5, 4, 3, 2, 1)==5 permutations

(6, 5, 4, 3, 2)==15 permutations.

(7, 6, 5, 4, 3) ==35 permutations.

(8, 7, 6, 5, 4) ==70 permutations.

(9, 8, 7, 6, 5) ==126 permutations.

{1  +  5  +  15  +  35  +  70  +  126} ==252 permutations

105 ==1  x  3  x  5  x  7

1357 ==1 combination ==4! ==24 permutations - or 24 arrangements of 1357.

XY = 6.

Labeling the parallelogram in this fashion.  AB = 30  BC = 20  CD = 30  DA = 20  AC = 40.

One way to solve this problem is to use the Law of Cosines in this fashion:

1)  Find the size of angle(CAB) by using the Law of Cosines on triangle(CAB).

2)  Find the size of angle(DAC) by using the Law of Cosines on triangle(DAC).

3)  Add the results of steps 1) and 2) to find the size of angle(DAB).

4)  Find the length of side BD by using the Law of Cosines on triangle(DAB).

A hint

You first can notice that the x number is 5/2 times bigger then the x+2 number(x is a number in the sequence).

Hoped my "hint" helped.

Or just do the math until you got the 10th number.

Let  a₁  be the first term.                                      Let  d  be the common difference.

Then:  a₂  =  a₁ + d         a₃  =  a₁ + 2d         a₄  =  a₁ + 3d         a₅  =  a₁ + 4d

a₅ / a₃  =  3     --->     ( a₁ + 4d ) / ( a₁ + 2d )  =  3

                        --->     a₁ + 4d  =  3( a₁ + 2d )

                        --->     a₁ + 4d  =  3a₁ + 6d

                        --->          -2a₁  =  2d

                        --->             a₁  =  -d

a₄ / a₂  =  ( a₁ + 3d ) / ( a₁ + d )  

Substituting  -d  for  a₁:    a₄ / a₂  =  ( -d + 3d ) / ( -d + d )  =  2d / 0     <---     Impossitble!

The smallest possible difference is \(\color{brown}\boxed{1}\). The series could go: 1, 2, 3, 4, 5, 6, ... 89, 90, 91. 

8x^2 + 15x + c =  7x^2 -20x + 8           rearrange as

x^2 + 35x + (c - 8)   = 0

To have two real roots, the discriminant must be > 0

So

35^2  - 4(1) (c - 8)  >  0

1225 - 4c + 32  >  0

1257  - 4c > 0

1257 > 4c

4c < 1257

c < 1257 / 4  =  314.5

So...any integer  c < 314.5  will work

And since c must be a positive integer  c =  1 * 2 * 3 * 4 * .....* 312 * 313 * 314

So....the product of  all integer c's  =  314!      [ a really BIG number !!!!!]